Let X1- X2 be two independent normal random variables with means - 1- - 2 and standard deviations - 1- - 2- respectively- Consider Y-X1-X2- - 1- 2-1- - 1-1- - 2-2- Then

∵ E(x_1)=1=E(x_2) and V(x_1)=1 and nbsp; nbsp; nbsp;V(x_2)=4 Hence y is also normally distributed with mean 0 and variance 5 ...

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Engineering Mathematics

Normal Distribution Properties

Let X1, X2 be...

Question

$Let X_{1}, X_{2}$ be two independent normal random variables with means $μ_{1}, μ_{2}$ and standard deviations $σ_{1}, σ_{2}, respectively. Consider Y = X_{1} - X_{2}, μ_{1} = μ_{2} = 1, σ_{1} = 1, σ_{2} = 2. Then$

Y is normally distributed with mean 0 and variance 1

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Y is normally distributed with mean 0 and variance 5

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

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

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Solution

The correct option is B Y is normally distributed with mean 0 and variance 5
$∵ E (x_{1}) = 1 = E (x_{2}) and V (x_{1}) = 1$

and $V (x_{2}) = 4$

Hence y is also normally distributed with mean 0 and variance 5

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Q. In the figure shown

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Let X1, X2 be two independent normal random variables with means μ1, μ2 and standard deviations σ1, σ2, respectively. Consider Y=X1−X2, μ1=μ2=1, σ1=1, σ2=2. Then

The correct option is B Y is normally distributed with mean 0 and variance 5∵ E(x1)=1=E(x2) and V(x1)=1 and V(x2)=4 Hence y is also normally distributed with mean 0 and variance 5

$Let X_{1}, X_{2}$ be two independent normal random variables with means $μ_{1}, μ_{2}$ and standard deviations $σ_{1}, σ_{2}, respectively. Consider Y = X_{1} - X_{2}, μ_{1} = μ_{2} = 1, σ_{1} = 1, σ_{2} = 2. Then$

The correct option is B Y is normally distributed with mean 0 and variance 5
$∵ E (x_{1}) = 1 = E (x_{2}) and V (x_{1}) = 1$

and $V (x_{2}) = 4$

Hence y is also normally distributed with mean 0 and variance 5