Let $x_{1}, x_{2}, . . . . . x_{n}$ be values taken by a variable $X$ and $y_{1}, y_{2}, . . . . . y_{n}$ be the values taken by variable $Y$ such that $y_{i} = {a x}_{i} + b; i = 1, 2, . . . . n$ . Then,

V a r (Y) = a^{2} V a r (X)

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V a r (X) = a^{2} V a r (Y)

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V a r (X) = V a r (X) + b

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none of these

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Solution

The correct option is A $V a r (Y) = a^{2} V a r (X)$
$V a r (Y) = V a r (a x_{i} + b) = E [y_{i} - ¯ y]^{2} = E [a x_{i} + b - (a ¯ x + b)]^{2} = E [a (x_{i} - ¯ x)^{2}] = a^{2} . E [(x_{i} - ¯ x)^{2}] = a^{2} . V a r (X)$

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Q. If

x_{1}, x_{2}, . . . . . x_{n}

are

n

values of a variable

X

and

y_{1}, y_{2}, . . . . . y_{n}

are

n

values of a variable

Y

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Let x1,x2,.....xn be values taken by a variable X and y1,y2,.....yn be the values taken by variable Y such that yi=axi+b;i=1,2,....n. Then,

The correct option is A Var(Y)=a2Var(X)Var(Y)=Var(axi+b)=E[yi−¯y]2=E[axi+b−(a¯x+b)]2=E[a(xi−¯x)2]=a2.E[(xi−¯x)2]=a2.Var(X)

Let $x_{1}, x_{2}, . . . . . x_{n}$ be values taken by a variable $X$ and $y_{1}, y_{2}, . . . . . y_{n}$ be the values taken by variable $Y$ such that $y_{i} = {a x}_{i} + b; i = 1, 2, . . . . n$ . Then,

The correct option is A $V a r (Y) = a^{2} V a r (X)$
$V a r (Y) = V a r (a x_{i} + b) = E [y_{i} - ¯ y]^{2} = E [a x_{i} + b - (a ¯ x + b)]^{2} = E [a (x_{i} - ¯ x)^{2}] = a^{2} . E [(x_{i} - ¯ x)^{2}] = a^{2} . V a r (X)$