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Let X be a  $N(\mu , \sigma^2)$ random variable and let $Y = \alpha X+\beta$, with $\alpha$ > $0$. How is $Y$ distributed?

$Y = \alpha X + \beta$

Mean = $\mu$

Standard Deviation = $\sigma$

Variance = $\sigma ^2$

Let Z be a random variable such that

$Z = \alpha X$

So Z is scaled by some $\alpha > 0$

$\mu _ Z = \alpha \mu$

$\sigma _Z = \alpha \sigma$

$Z \sim$ $N(\mu _ Z , \sigma _Z ^2 ) \sim N(\alpha \mu, (\alpha \sigma)^2)$

Now,

$Y = Z + \beta$

So the distribution will be shifted depending upon the magnitude and sign of $\beta$,

But it doesn't change the standard deviation but it does change the mean.

$\mu_y = \mu_z + \beta = \alpha \mu + \beta$

Hence,

$Y \sim N(\alpha \mu + \beta, (\alpha \sigma)^2)$

### 1 comment

what about mode , median here ?? they also change ...

$E(Y)=E(\alpha x+\beta )=E(\alpha x)+E(\beta )=\alpha E(x)+\beta =\alpha \mu +\beta$

Since E(ax)=aE(x) and E(constant)=constant

$Var(Y)=Var(\alpha x+\beta )=Var(\alpha x)+Var(\beta )=\alpha^{2} Var(x)+\beta =\alpha^{2} \sigma ^{2}$

Since Var(ax)=$a^{2}$Var(x) and Var(constant)=0

Hence, Y is a N($\alpha \mu +\beta$,  $\alpha^{2} \sigma ^{2}$) random variable

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