Theoretical DistributionsMCQPYQ Sep 24Question 3559 of 230
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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 (X+Y)\displaystyle (X + Y) is normally distributed with:

Options

AMean =22\displaystyle = 22 and S.D =25\displaystyle = 25
BMean =22\displaystyle = 22 and S.D =7\displaystyle = 7
CMean =22\displaystyle = 22 and S.D =5\displaystyle = 5
DMean =22\displaystyle = 22 and S.D =49\displaystyle = 49
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Correct Answer

✅ Option c — Mean =22\displaystyle = 22 and S.D =5\displaystyle = 5

All Options:

  • AMean =22\displaystyle = 22 and S.D =25\displaystyle = 25
  • BMean =22\displaystyle = 22 and S.D =7\displaystyle = 7
  • CMean =22\displaystyle = 22 and S.D =5\displaystyle = 5
  • DMean =22\displaystyle = 22 and S.D =49\displaystyle = 49

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Detailed Solution & Explanation

**Distribution of the Sum of Two Independent Normal Variables** Let X\displaystyle X and Y\displaystyle Y be independent normal random variables: - X∼N(μX,σX2)\displaystyle X \sim N(\mu_X, \sigma_X^2) with μX=10\displaystyle \mu_X = 10 and σX=3\displaystyle \sigma_X = 3 - Y∼N(μY,σY2)\displaystyle Y \sim N(\mu_Y, \sigma_Y^2) with μY=12\displaystyle \mu_Y = 12 and σY=4\displaystyle \sigma_Y = 4 Let Z=X+Y\displaystyle Z = X + Y. The sum Z\displaystyle Z is also normally distributed: Z∼N(μZ,σZ2)Z \sim N(\mu_Z, \sigma_Z^2) **Step 1: Calculate the Mean of Z\displaystyle Z** μZ=E[X+Y]=E[X]+E[Y]=10+12=22\mu_Z = E[X + Y] = E[X] + E[Y] = 10 + 12 = 22 **Step 2: Calculate the Variance of Z\displaystyle Z** Since X\displaystyle X and Y\displaystyle Y are independent: σZ2=Var(X)+Var(Y)=σX2+σY2\sigma_Z^2 = \text{Var}(X) + \text{Var}(Y) = \sigma_X^2 + \sigma_Y^2 σZ2=32+42=9+16=25\sigma_Z^2 = 3^2 + 4^2 = 9 + 16 = 25 **Step 3: Calculate the Standard Deviation of Z\displaystyle Z** σZ=σZ2=25=5\sigma_Z = \sqrt{\sigma_Z^2} = \sqrt{25} = 5 Thus, X+Y\displaystyle X + Y is normally distributed with a mean of 22\displaystyle 22 and standard deviation of 5\displaystyle 5. Hence, **Option C** 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.

Key Concepts to Understand

Standard Deviation

A measure of the dispersion or spread of values in a data set around the mean. It is the square root of Variance. A lower standard deviation indicates data points are clustered close to the mean; a higher value indicates greater spread.

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