Q.4.3 - A First Course in Probability

Question

If X has distribution function F, what is the distribution function of the random variable αX + β, where α and β are constants,  $\alpha \ne 0?$

Step-by-Step Solution

Answer

In the given information the answer is $F_{\alpha X+\beta }\left(y\right)=F\left(\frac{y-\beta }{\alpha }\right)$

Step 1:Given Information

X has distribution function F.

The distribution function F at X =x has the property that represents the probability that X is at most y.

$F\left(x\right)=P(X\le x)$

Step 2 : Calculation

Let us determine the cumulative distribution function of $\alpha X+\beta (\alpha \ne 0)$.

$F_{\alpha X+\beta }\left(y\right)=P(\alpha X+\beta \le y)$

                $=P(\alpha X\le y-\beta )$

                 $=P\left(X\le \frac{y-\beta }{\alpha }\right)$

                 $=F\left(\frac{y-\beta }{\alpha }\right)$

Step 3:Final Answer

The final answer is $F_{\alpha X+\beta }\left(y\right)=F\left(\frac{y-\beta }{\alpha }\right)$