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 $Y$ such that $y_{i} = \frac{x_{i} - a}{h}; i = 1, 2, . . . . ., n$ , then

V a r (Y) = V a r (X)

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

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

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

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Solution

The correct option is B $V a r (X) = h^{2} V a r (Y)$
$V a r (Y) = E [y_{i} - ¯ y]^{2} = E {[\frac{x_{i} - a}{h} - \frac{¯ x - a}{h}]}^{2} = E {[\frac{x_{i}}{h} - \frac{¯ x}{h}]}^{2} = E {[\frac{1}{h} (x_{i} - ¯ x)]}_{i}^{2} = \frac{1}{h^{2}} E [x - ¯ x]^{2} = \frac{1}{h^{2}} V a r (X)$

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If x1,x2,.....xn are n values of a variable X and y1,y2,.....yn are n values of a variable Y such that yi=xi−ah;i=1,2,.....,n, then

The correct option is B Var(X)=h2Var(Y)Var(Y)=E[yi−¯y]2=E[xi−ah−¯x−ah]2=E[xih−¯xh]2=E[1h(xi−¯x)]2=1h2E[x−¯x]2=1h2Var(X)

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 $Y$ such that $y_{i} = \frac{x_{i} - a}{h}; i = 1, 2, . . . . ., n$ , then

The correct option is B $V a r (X) = h^{2} V a r (Y)$
$V a r (Y) = E [y_{i} - ¯ y]^{2} = E {[\frac{x_{i} - a}{h} - \frac{¯ x - a}{h}]}^{2} = E {[\frac{x_{i}}{h} - \frac{¯ x}{h}]}^{2} = E {[\frac{1}{h} (x_{i} - ¯ x)]}_{i}^{2} = \frac{1}{h^{2}} E [x - ¯ x]^{2} = \frac{1}{h^{2}} V a r (X)$