Let x_{1}, x_{2} be two independent normal random variables with means \mu_{1}, \mu_{2} and standard deviations \sigma_{1}, \sigma_{2} respectively. Consider Y=X_{1}-X_{2} ; \mu_{1}=\mu_{2}=1, \sigma_{1}=1 ; \sigma_{2}=2, Then,

Correct Answer is : Y is normally distributed with mean 0 and variance 5

Y is normal distributed with mean 0 and variance 1

{Y} is normally distributed with mean 0 and variance 5

Y has mean 0 and variance 5, but is NOT normally distributed

Y has mean 0 and variance 1, but is NOT normally distributed

GATE Mechanical Engineering (ME) 2018 Shift 1 Solved Paper

Question : 46 of 65

Marks: +1, -0

Let

x_{1}, x_{2}

be two independent normal random variables with means

µ_{1}, µ_{2}

and standard deviations

σ_{1}, σ_{2}

respectively. Consider

Y=X_{1}-X_{2} ; µ_{1}=µ_{2}=1, σ_{1}=1 ; σ_{2}=2

, Then,

Solution:

µ_{1}=1, µ_{2}=1, σ_{1}=1, σ_{2}=2

x_{1}

and

x_{2}

are two independent random variables

Y=X_{1}-X_{2}

µ(Y)=µ(X_{1}-X_{2})

=µ(X_{1})-µ(X_{2})=µ_{1}-µ_{2}=1-1=0

Var(Y)=Var(X_{1}-X_{2})

=Var(X_{1})+Var(X_{2})-Cov(X_{1}, X_{2})

Since

X_{1}

and

X_{2}

are independent variables

Var(Y)=Var(X_{1})+Var(X_{2})

=σ_{1}^{2}+σ_{2}^{2}=1+4

Var(Y)=5

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