Theoretical Distributions

If the first quartile ($\displaystyle Q_1$) and third quartile ($\displaystyle Q_3$) of a normal distribution are 22 and 28, what is the median of the distribution?

The Mode of binomial distribution $\displaystyle B(7, \frac{1}{3})$ is

If $\displaystyle p$ is increased for a fixed $\displaystyle n$; the Binomial distribution shifts to the

The variance of a binomial distribution with parameters $\displaystyle n$ and $\displaystyle p$ is:

An example of a bi-parametric discrete probability distribution is

Probability distribution may be

The mean of the Binomial distribution $\displaystyle B(4, 1/3)$ is equal to

The probability that a student is not a swimmer is $\displaystyle 1/5$ then the probability that out of five students four are swimmer is

If mean and variance are $\displaystyle 5$ and $\displaystyle 3$ respectively then relation between $\displaystyle p$ and $\displaystyle q$ is:

Find mode when $\displaystyle n = 15$ and $\displaystyle p = 1/4$ in binomial distribution?

In a binomial distribution $\displaystyle B(n, p)$, $\displaystyle n = 4$, $\displaystyle P(X=2) = 3P(X=3)$ find $\displaystyle p$

If the probability of success in a binomial distribution is less than one-half, then the binomial distribution

A coin with probability for heads as $\displaystyle 1/5$ is tossed $\displaystyle 100$ times. The standard deviation of the number of head $\displaystyle 5$ turned up is.

If $\displaystyle X$ is a binomial variate with $\displaystyle p = 1/3$, for the experiment of $\displaystyle 90$ trials, then the standard deviation is equal to:

Four unbiased coins are tossed simultaneously. The expected no. of heads is:

For a binomial distribution, there may be -

The standard deviation of binomial distribution is:

The incidence of skin diseases in a chemical plant occurs in such a way that the workers have $\displaystyle 20\%$ chance of suffering from it. What is the probability that out of $\displaystyle 6$ workers $\displaystyle 4$ or more will have skin diseases?

If mean and variance of a random variable which follows the Binomial Distribution are $\displaystyle 7$ and $\displaystyle 6$ respectively, then the probability of success is:

If six coins are tossed simultaneously. The probability of obtaining exactly two heads are:

For a binomial distribution the mean and standard deviation are $\displaystyle 10$ and $\displaystyle 3$ respectively. Find the value of $\displaystyle n$.

For a binomial distribution, the variance is $\displaystyle 0.2$ and the mean is $\displaystyle 0.6$. The probability of getting $\displaystyle 3$ successes out of a trial of $\displaystyle 5$ is _____.

When '$\displaystyle p$' $\displaystyle = 0.5$, the

If mean and standard deviation of a binomial distribution is $\displaystyle 10$ and $\displaystyle 4$ respectively, $\displaystyle p$ will be

The mean of Binomial distribution is $\displaystyle 4$ and the Standard Deviation $\displaystyle \sqrt{3}$, what is the value of $\displaystyle p$.

The mean of binomial distribution is

In Binomial distribution the trails are

A binomial distribution is

The maximum value of the variance of binomial distribution with parameters $\displaystyle n$ and $\displaystyle p$ is

The max. value of the variance of binomial distribution with parameters $\displaystyle n$ and $\displaystyle p$ is

If $\displaystyle X$ & $\displaystyle Y$ are two independent variables such that $\displaystyle X \sim B(n_1, p)$ and $\displaystyle Y \sim B(n_2, p)$ then the parameter of $\displaystyle Z = X+Y$ is

Five coins tossed 3200 times. The number of times 5 heads appeared is.

Find the probability of a success for the binomial distribution satisfying the following relation $\displaystyle 4 P(x=4) = P(x=2)$ and having the parameter $\displaystyle n$ as six.

An experiment succeeds thrice as after it fails. If the experiment is repeated 5times, what is the probability of having no success at all?

The overall percentage of failures in a certain examination was 30. What is the probability that out of a group 6 candidates at least four passed the examination? Note: Exact Ans is 0.74431

What is the probability of getting exactly 2 head in 7 tosses of a fair coin?

The Binomial Distribution for which mean = 15 and variance = 6.0 is

The SD of a binomial distribution with parameter $\displaystyle n$ and $\displaystyle p$ is

Bivariate Data are the data collected for Note: From correlation regression chapter.

If $\displaystyle x$ is binomial variate with parameter 15 and $\displaystyle 1/3$ what is the value of mode of the distribution.

The mean of a binomial distribution with parameter $\displaystyle n$ and $\displaystyle p$ is

The Binomial distribution $\displaystyle n=9$ and $\displaystyle p=1/3$. What is the value of the variance?

In a Binomial Distribution $\displaystyle B(n, p)$, $\displaystyle n=4$, then $\displaystyle P(x=2) = 3 P(x=3)$. Find $\displaystyle p$

The variance of a binomial distribution with parameters $\displaystyle n$ and $\displaystyle p$ is

What is the probability of getting 3 heads if 6 unbiased coins are tossed simultaneously?

The mode of the binomial distribution for which the mean is 4 variance 3 is equal to ?

If a variate $\displaystyle x$ has, mean=variance, then the distribution will be _____

In a Binomial distribution $\displaystyle n=9$ and $\displaystyle p=1/3$. What is the value of Variance.

Examine the validity of the following: Mean and standard deviation of a binomial distribution are 10 and 4 respective:

The probability of a man hitting the target is $\displaystyle 1/4$. If he fires 7 times, the probability of hitting the target at least twice is :

If mean and variance are 5 and 3 respectively then relation between $\displaystyle p$ and $\displaystyle q$ is

If a coin is tossed 5 times then the prob. of getting Tail and Head occurs alternatively is:

The probability that a student is not a swimmer is $\displaystyle 1/5$, then the probability that out of five students four are swimmers is:

A random variable $\displaystyle x$ follows Binomial Distribution With $\displaystyle E(x)=2$ and $\displaystyle V(x)=1.2$. then the value of $\displaystyle n$ is

If $\displaystyle x$ is binomial with parameter $\displaystyle 15$ and $\displaystyle \frac{1}{3}$, what is mode of the distribution?

When '$\displaystyle p$' is large than $\displaystyle 0.5$, the Binomial Distribution is

A die is thrown $\displaystyle 100$ times, if getting an even number is considered a success then the variance number of success.

The standard deviation of Binomial distribution is

In Binomial distribution $\displaystyle n=9$ and $\displaystyle P=\frac{1}{3}$, what is the value of variance?

The overall percentage of failure in a certain examination is $\displaystyle 0.30$. What is the probability that out of a group of $\displaystyle 6$ candidates at least $\displaystyle 4$ passed the examination?

For binomial distribution $\displaystyle E(x)=2$, $\displaystyle V(x)=\frac{4}{3}$. Find the value of $\displaystyle n$.

Parameter is a characteristic of:

If mean & variance are $\displaystyle 5$ and $\displaystyle 3$ respectively then relation between $\displaystyle p$ and $\displaystyle q$ is:

Find the variance of binomial distribution with $\displaystyle n=10$, $\displaystyle p=0.3$

When $\displaystyle p=0.5$, the binomial distribution is

If mean and standard deviation of a binomial distribution is $\displaystyle 10$ and $\displaystyle 2$ respectively, $\displaystyle q$ will be

A random variable $\displaystyle X$ follows Binomial Distribution With $\displaystyle E(x)=2$ and $\displaystyle V(x)=1.2$, then the value of $\displaystyle n$ is

When '$\displaystyle p$' is large than $\displaystyle 0.5$, the Binomial Distribution is:

If $\displaystyle X$ is a binomial variable with parameters $\displaystyle n$ and $\displaystyle p$, then $\displaystyle X$ can assume

$\displaystyle X$ is a binomial variable such that $\displaystyle 2P(X=2)=P(X=3)$ and Mean of $\displaystyle X$ is known to be $\displaystyle \frac{10}{3}$. What would be the probability that $\displaystyle X$ assumes at most the value $\displaystyle 2$?

In a Poisson distribution if $\displaystyle P(X=4)=P(X=5)$ then the parameter of Poisson distribution is:

For a poisson distribution:

In Poisson distribution, if $\displaystyle P(X=2) = \frac{1}{2}P(X=3)$ find $\displaystyle m$?

Which of the following is uni-parametric distribution?

Which one of the following has Poisson distribution?

For a Poisson distributed variable $\displaystyle X$, we have $\displaystyle P(X=7)=8P(X=9)$, the mean of the distribution is:

If the parameter of Poisson distribution is $\displaystyle m$ and $\displaystyle (Mean + S.D.) = \frac{6}{25}$ then find $\displaystyle m$:

$\displaystyle X$ is a Poisson variate satisfying the following condition $\displaystyle 9P(X=4)+30P(X=6)=P(X=2)$. What is the value of $\displaystyle P(X \le 1)$?

For a Poisson variate $\displaystyle X$, $\displaystyle P(X=2)=3P(X=4)$, then the standard deviation of $\displaystyle X$ is

$\displaystyle 4$ coins were tossed $\displaystyle 1600$ times. What is the probability that all $\displaystyle 4$ coins do not turn head upward at a time?

If $\displaystyle X$ is a Poisson variable and $\displaystyle P(X=1) = P(X=2)$, then $\displaystyle P(X=4)$ is

Which one of the following is an unparametric distribution?

It is Poisson variate such that $\displaystyle P(X=1) = 0.7, P(X=2) = 0.3$, then $\displaystyle P(X=0) =$

The average number of advertisements per page appearing in a newspaper is 3. What is the probability that in a particular page zero number of advertisements are there?

If, for a Poisson distributed random variable $\displaystyle X$, the probability for $\displaystyle X$ taking value 2 is 3 times the probability for $\displaystyle X$ taking value 4, then the variance of $\displaystyle X$ is

The manufacturer of a certain electronic component is certain that $\displaystyle 2\%$ of his product is defective. He sells the components in boxes of 120 and guarantees that not more than $\displaystyle 2\%$ in any box will be defective. Find the probability that a box selected at random would fail to meet the guarantee? ($\displaystyle e^{-2.4} = 0.0907$)

A renowned hospital usually admits 200 patients everyday. One percent patients, on an average, require special room facilities. On one particular morning, it was found that only one special room is available. What is the probability that more than 3 patients would require special room facilities?

If Standard Deviation is 1.732 then what is the value of poisson distribution. The $\displaystyle P(-2.48 < X < 3.54)$ is

If a Poisson distribution is such that $\displaystyle P(X=2) = P(X=3)$ then the variance of the distribution

Between 9 AM and 10 AM, the average number of phone calls per minute coming into the switchboard of a company is 4. Find the probability that in one particular minute there will be either 2 phone calls or no phone calls (given $\displaystyle e^{-4} = 0.018316$)

If a Poisson distribution is such that $\displaystyle P(X=2) = P(X=3)$, then the standard deviation of the distribution is:

The mean of Poisson distribution is 4. The probability of two-successes is____.

A company produces 5 defective items out of 300 items. The probability distribution follows

If a random variable $\displaystyle X$ follows Poisson distribution such that $\displaystyle P(X=1) = P(X=2)$, then the mean of the distribution is:

If for a normal distribution $\displaystyle Q_1 = 54.52$ and $\displaystyle Q_3 = 78.86$, then the median of the distribution is

The number of accidents in a year attributed to taxi drivers in a locality follows Poisson distribution with average 2. Out of 500 taxi drivers of that area, what is the number of drivers with at least 3 accidents in a year? (Given that $\displaystyle e^{-2} = 0.18$)

Which one is uniparametric distribution?

Find the probability that at least 2 defective bolts will be found in a box of 200 bolts. If it is known that $\displaystyle 2\%$ of such bolts are expected to be defective (Given: $\displaystyle e^{-4} = 0.0183$)

Number of misprints per page of a thick book follows

If for a Poisson variable $\displaystyle X$, $\displaystyle E(X^2) = 3E(X)$, what is the variance of $\displaystyle X$?

If $\displaystyle P(X=2) = P(X=3)$ for a Poisson Variate $\displaystyle X$, then $\displaystyle E(X)$ is

In Poisson distribution which of them is same.

Number of defects in clothes a garments showroom will form a

In a certain Poisson frequency distribution, the probability corresponding to two success is half the probability corresponding to three successes. The mean of the distribution is

Which one is not a condition of Poisson model

In ______ distribution, mean = variance.

For a Poisson variate X, $\displaystyle P(X=1) = P(X=2)$. What is the mean of X?

For a Poisson distribution,

For Poisson Distribution :

For a Poisson variate X, $\displaystyle P(X=2) = 3 P(X=4)$. Then the standard deviation of X is

If X be a Poisson variates with parameter $\displaystyle \lambda$, then find $\displaystyle P(3 < X < 5)$ (Given $\displaystyle e^{-0.36783} = 0.6923$)

In a Poisson Distribution $\displaystyle P(X=0) = P(X=2)$. Find $\displaystyle E(X)$

Name of the distribution which has Mean=Variance

If $\displaystyle 5\%$ of the electric bulbs manufactured by a company are defective, use Poisson distribution to find the probability that in a sample of $\displaystyle 100$ bulbs, $\displaystyle 5$ bulbs will be defective. [Given : $\displaystyle e^{-5} = 0.007$ ]

For a Poisson variate X, $\displaystyle P(X=1) = P(X=2)$, what is the mean of X?

In a Poisson distribution if $\displaystyle P(X=4) = P(X=5)$ then the parameter of Poisson distribution is:

If Poisson distribution is such that $\displaystyle P(X=2) = P(X=3)$ then the Standard Deviation of the distribution is

To find the distribution of number of airplanes crashing every hour in the world, which of the following distribution is appropriate to apply:

The mean and variance are equal for which of the following:

For the Poisson distribution:

Which of the following is uni-parametric distribution

The probability that a man aged $\displaystyle 45$ years will die within a year is $\displaystyle 0.012$. What is the probability that of $\displaystyle 10$ men, at least $\displaystyle 9$ will reach their $\displaystyle 46$th birthday? [Given $\displaystyle e^{-0.12} = 0.88692$ ]

In a certain manufacturing process, $\displaystyle 5\%$ of the tools produced turn out to be defective. Find the probability that in a sample of $\displaystyle 40$ tools, at most $\displaystyle 2$ will be defective: Given $\displaystyle e^{-2} = 0.135$

If standard deviation of a Poisson distribution is $\displaystyle 2$, then its Mode

In Poisson distribution if $\displaystyle P(X=4) = P(X=5)$ then the parameter of Poisson distribution is:

In _________ distribution, mean = variance.

The mean of Poisson distribution is $\displaystyle 4$. The probability of two-successes in

What is the first quartile of $\displaystyle x$ having the following probability density function? $\displaystyle f(x) = \frac{1}{\sqrt{72\pi}} e^{-\frac{(x-10)^2}{72}}$ for $\displaystyle -\infty < x < \infty$

If the area of standard normal curve between $\displaystyle z=0$ to $\displaystyle z=1$ is $\displaystyle 0.3412$, then the value of $\displaystyle \phi(1)$ is

What is the mean of X having the following density function? $\displaystyle f(x) = \frac{1}{\sqrt{4\pi}2} e^{-\frac{(x-10)^2}{32}}$ for $\displaystyle -\infty < x < \infty$

Area between $\displaystyle -1.96$ to $\displaystyle +1.96$ in a normal distribution is:

If the points of inflexion of a normal curve are 40 & 60 respectively, then mean deviation is:

Area under $\displaystyle \pm 3\sigma$

What is the mean and SD if $\displaystyle f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{(x-3)^2}{2}}$, $\displaystyle -\infty < x < \infty$.

If we change the parameter(s) of a distribution the shape of probability curve does not change.

The quartile deviation of a normal distribution with mean 10 and standard deviation 4 is

For a normal distribution, the value of third moment about mean is.

In normal distribution, Mean, Median and Mode are:

For a certain type of mobile, the length of time between charges of the battery is normally distributed with a mean of 50 hours and a standard deviation of 15 hours. A person owns one of these mobiles and want to know the probability that the length of time will be between 50 and 70 hours is (given $\displaystyle q(1.33) = 0.9082, q(0) = 0.5$)?

Let $\displaystyle x$ be normal distribution with mean 2.5 and variance 1. If $\displaystyle P(x \le 2.5) = 0.4772$ and that the cumulative normal probability value at 2 is 0.9772, then $\displaystyle a = ?$

In a normal distribution, variance is 16 then the value of mean deviation is.

Skewness of Normal Distribution is:

The speeds of a number of bikes follow a normal distribution model with a mean of 83 km/hr and a standard deviation of 9.4 km/hr. Find the probability that a bike picked at random is travelling at more than 95km/hr.? Given $\displaystyle P(Z > 1.28) = 0.1003$

In a Standard Normal distribution, then the value of the mean($\displaystyle \mu$) and SD ($\displaystyle \sigma$) is:

If 'x' and 'y' are independent normal variate with mean and SD $\displaystyle \mu_1, \mu_2$ and $\displaystyle \sigma_1, \sigma_2$ respectively, then for $\displaystyle z = x + y$ which also follows normal distribution mean and SD are:

For a normal distribution, the ratio of mean deviation to the standard deviation is _____

In a class of 100 students, the mean marks was 50 with Standard Deviation 14.9. Assuming the distribution of marks to be normal, find the number of students who obtained more than 70% marks (at $\displaystyle Z = 1.34$, area $\displaystyle = 0.4099$).

Quartile deviation of a normal distribution with Mean of 10 and SD of 4 is:

If X and Y are 2 independent normal variables with mean as 10 and 12 and Standard Deviation (S.D) as 3 and 4 respectively, then $\displaystyle (X + Y)$ is normally distributed with:

If the quartile deviation of a normal curve is 4.05, then its mean deviation is

In a normal distribution skewness is _____

The points of inflexion of the normal curve $\displaystyle f(t) = \frac{1}{4\sqrt{2\pi}} e^{-\frac{(t-32)^2}{32}}$ are

If X and Y are independent Normal Variables with mean 100 and 80 respectively and standard deviation as 4 and 3 respectively. What is the distribution of $\displaystyle (X+Y)$?

If X is normal variate with mean 6 and variance 16 then the value of the probability. $\displaystyle P(2 \le X \le 10)$ is equal to.

The total area of the normal curve is

If the mean deviation of a normal variable is 16, what is its quartile deviation?

For Poisson fitting to an observed frequency distribution

The mean deviation about median of a standard normal variate is

If the points of inflexion of a normal curve are 40 and 60 respectively, then its mean deviation is

What is the first quartile of X having the following probability density function? $\displaystyle f(t) = \frac{1}{4\sqrt{2\pi}} e^{-\frac{(t-10)^2}{32}}$ for $\displaystyle -\infty < X < \infty$

For the normal distribution density function $\displaystyle f(x) = K e^{-\frac{(x-6)^2}{8}}$, the mean and variance are.

The mean deviation of normal distribution is 16. The Quartile Deviation is

The Quartile Deviation of the normal distribution $\displaystyle f(x) = \frac{1}{\sqrt{18\pi}} e^{-\frac{(x-10)^2}{18}}$, $\displaystyle -\infty < x < \infty$ is

If X and Y are two independent normal random distributions with mean and SD's are (10, 5) and (15, 12) these mean and SD of (x+y) is.

If the two quartiles of a normal distribution are 47.30 and 52.70 respectively, what is the mode of the distribution? Also find the mean deviation about median of this distribution.

X follows normal distribution with mean as 50 and variance as 100. What is $\displaystyle P(X \ge 60)$? [Given $\displaystyle P(1) = 0.8413$]

The mean and mode of the normal distribution

Area covered normal curve by $\displaystyle \mu \pm 3\sigma$

The Quartile Deviation of Normal Distribution with mean is 10 and variance is 16 is

For a normal distribution with mean 150 and SD is 45, Find Q1 and Q3

The normal curve is

For a normal distribution $\displaystyle Q_1 = 54.32$ and $\displaystyle Q_3 = 78.86$, then the median of the distribution is

What is the mean of X having the following density function $\displaystyle f(x) = \frac{1}{4\sqrt{2\pi}} e^{-\frac{(x-10)^2}{32}}$ for $\displaystyle -\infty < x < \infty$

What is the first quartile of X having the following probability density function? $\displaystyle F(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{(x-10)^2}{32}}$ for $\displaystyle -\infty < x < \infty$

If X follows normal distribution with $\displaystyle \mu = 50$ and $\displaystyle \sigma = 10$, what is value of $\displaystyle P(X \le 60 | X > 50)$?

For a normal distribution with mean as 500 and SD as 120, what is the value of k so that the interval [500, k] covers 40.32 per cent area of the normal curve? (Given $\displaystyle \phi(1.30) = 0.9032$)

An example of a bi-parametric continuous probability distribution

What is the mean of X having the following density function? $\displaystyle f(x) = \frac{1}{4\sqrt{2\pi}} e^{-\frac{(x-10)^2}{32}}$ for $\displaystyle -\infty < x < \infty$

The variance of standard normal distribution

For a normal distribution, the first and third quartile are given to be 37 and 49, the mode of the distribution is.

Area between -1.96 to +1.96 in a normal distribution is :

The speeds of bikes follow a normal distribution model with a mean of $\displaystyle 80$ km/hr. and a standard deviation of $\displaystyle 9.4$ km/hr. Find the probability that a bike picked at random is travelling at more than $\displaystyle 95$ km/hr.? $\displaystyle [P(z) = P(1.60) = 0.4452]$

Which of the following is not a property of normal distribution?

For a continuous random variable following standard normal distribution, what is the value of standard deviation?

If the inflexion points of a normal distribution are $\displaystyle 6$ and $\displaystyle 14$. Find its SD

Normal distribution is also known as

The mean deviation about median of standard normal variate is

If the Quartile Deviation of a normal distribution with mean $\displaystyle 10$ and SD $\displaystyle 4$ is

If the two Quartiles $\displaystyle N(\mu, \sigma^2)$ are $\displaystyle 14.6$ and $\displaystyle 25.4$ respectively. What is the standard deviation of the distribution?

The wages of workers of a factory follows

If the inflexion points of a Normal Distribution are $\displaystyle 6$ and $\displaystyle 14$. Find its SD?

An approximate relation between quartile deviation (QD) and standard deviation (S.D.) of normal distribution is:

Which of the following is not a characteristic of a normal probability distribution?

The Interval $\displaystyle (\mu - 3\sigma, \mu + 3\sigma)$ covers

If the mean deviation of a normal variable is $\displaystyle 16$, what is its quartile deviation?

If the quartile deviation of a normal curve is $\displaystyle 4.05$, then its mean deviation is

In a normal distribution skewness is ______

The points of inflexion of the normal curve $\displaystyle f(t) = \frac{1}{4\sqrt{2\pi}}e^{-\frac{(t-10)^2}{32}}$ are

The mean deviation abut median of standard normal variate is

The probability density function of a normal variable $\displaystyle x$ is given by

The average weekly food expenditure of a group of families has a normal distribution with mean $\displaystyle ₹1,800$ and standard deviation $\displaystyle ₹300$. What is the probability that out of $\displaystyle 5$ families belonging to this group, at least one family has weekly food expenditure in excess of $\displaystyle ₹2,100$? Given $\displaystyle F(1) = 0.84$.

For a standard normal distribution, the points of inflexion are given by

The interval $\displaystyle \mu-3\sigma$ & $\displaystyle \mu+3\sigma$ covers

A random variable X has the following probability density function: $\displaystyle f(x) = 6x(1-x), 0 \le x \le 1$. Then the mean is

If X is a Poisson variable such that $\displaystyle P(X=1) = P(X=2)$ then the variance is

A random variable has the following probability distribution: X: [0, 1, 2, 3], P(X): [?, 1/3, 1/4, 1/5], sum of P(X) is 1. Find expected value of X.

What will be the mode of the Binomial distribution in which mean is 20 & Standard Deviation is $\displaystyle \sqrt{10}$ ?

If 3 percent of ceramic cup manufactured by a company are known to be defective. What is the probability that a sample of 100 cups are taken from the production process, of that company would contain exactly one defective cup? (Use $\displaystyle e^{-3}=0.0498$)

The mean deviation of normal distribution is approximately equal to

For a binomial distribution, if the mean is 10 and the standard deviation is 3, find n. (number of trials)

If a binomial distribution has $\displaystyle n = 25$ and $\displaystyle p = 0.2$ what is its mean?

If X is a Poisson variate such that $\displaystyle P(X = 1) = 0.3$ and $\displaystyle P(X = 2) = 0.2$ then $\displaystyle P(X = 0) =$

A quality control inspector finds that 20% of light bulbs are defective. If a batch of 5 light bulbs is tested, what is the probability that exactly 1 bulb is defective?

Poisson probability distribution is appropriately applied in

If the points of inflexion of a normal curve are 6 and 14, then standard deviation of the distribution is

What is the probability of making 3 corrected guesses in 5 True-False answer type questions?

If 5% of the families in large population city do not use gas as a fuel, what will be the probability of selecting 10 families in a random sample of 100 families who do not use gas as a fuel? [Given that $\displaystyle e^{-5} = 0.0067$]

An emergency room receives an average of 3 patients per hour. What is the probability that exactly 2 patients arrive in an hour? (Given: $\displaystyle e^0 = 1$, $\displaystyle e^{-1} = 0.367$, $\displaystyle e^{-2} = 0.135$, $\displaystyle e^{-3} = 0.049$, $\displaystyle e^{-4} = 0.018$, $\displaystyle e^{-5} = 0.0067$)

If a binomial distribution has $\displaystyle n = 20$ and $\displaystyle p = 0.3$ what is its variance?

It is given that $\displaystyle X$ has normal distribution with mean zero and standard deviation one. Also given that $\displaystyle P[-2 < X < 2] = 0.95$, $\displaystyle P[-2 < X < -1.5] = 0.045$. Find the probability for $\displaystyle P[0 < x < 1.5]$.

The probability mass function of a distribution is given below in a tabular form: $\displaystyle p(x)$ is $\displaystyle k$ at $\displaystyle x=0$, $\displaystyle 2k+k^2$ at $\displaystyle x=1$, $\displaystyle 3k$ at $\displaystyle x=2$, $\displaystyle 2k+k^2$ at $\displaystyle x=3$, and $\displaystyle k$ at $\displaystyle x=4$. Where $\displaystyle k$ is a non-negative constant. The median of the distribution is