Theoretical DistributionsPYQ Sept 25Question 4188 of 230
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It is given that X\displaystyle X has normal distribution with mean zero and standard deviation one. Also given that P[−2<X<2]=0.95\displaystyle P[-2 < X < 2] = 0.95, P[−2<X<−1.5]=0.045\displaystyle P[-2 < X < -1.5] = 0.045. Find the probability for P[0<x<1.5]\displaystyle P[0 < x < 1.5].

Options

A0.63
B0.53
C0.33
D0.43
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Correct Answer

✅ Option d — 0.43

All Options:

  • A0.63
  • B0.53
  • C0.33
  • D0.43

Detailed Solution & Explanation

Let X\displaystyle X be a standard normal variable, X∼N(0,1)\displaystyle X \sim N(0, 1). The probability density curve of the standard normal distribution is symmetric about 0\displaystyle 0.
1. **Using symmetry property**:
Since the distribution is symmetric about the mean 0\displaystyle 0, we have:
P[−2<X<2]=0.95  ⟹  P[0<X<2]=0.952=0.475P[-2 < X < 2] = 0.95 \implies P[0 < X < 2] = \frac{0.95}{2} = 0.475
2. **Symmetry of intervals**:
The probability of X\displaystyle X lying in the interval [−2,−1.5]\displaystyle [-2, -1.5] is equal to the probability of X\displaystyle X lying in the interval [1.5,2]\displaystyle [1.5, 2]:
P[1.5<X<2]=P[−2<X<−1.5]=0.045P[1.5 < X < 2] = P[-2 < X < -1.5] = 0.045
3. **Partitioning the interval [0,2]\displaystyle [0, 2]**:
We can divide the interval [0,2]\displaystyle [0, 2] into two non-overlapping intervals [0,1.5]\displaystyle [0, 1.5] and [1.5,2]\displaystyle [1.5, 2]:
P[0<X<2]=P[0<X<1.5]+P[1.5<X<2]P[0 < X < 2] = P[0 < X < 1.5] + P[1.5 < X < 2]
Substitute the known values:
0.475=P[0<X<1.5]+0.0450.475 = P[0 < X < 1.5] + 0.045
P[0<X<1.5]=0.475−0.045=0.43P[0 < X < 1.5] = 0.475 - 0.045 = 0.43
Therefore, P[0<X<1.5]=0.43\displaystyle P[0 < X < 1.5] = 0.43.
Hence, **Option D** is the correct answer.

About This Chapter: Theoretical Distributions

Paper

Paper 3: Quantitative Aptitude

Weightage

4-6 Marks

Key Topics

Binomial, Poisson, Normal Distribution

This chapter covers Binomial, Poisson, Normal Distribution and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 4-6 Marks weightage. Focus on understanding core concepts rather than memorizing.

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