Question

 If X has uniform distribution over [0,1], find the density function and distribution function of Y D 3X: Identify the distribution of Y

Answer 1

Solution:

$X\sim U[0,1]$

$Y=3X \\\Rightarrow X=\dfrac Y3$

$P(Y\le r)=P[0\le X\le \dfrac r3]=\dfrac r3$ ...(i)

As $0<x<1, 0<\dfrac Y3<1$

$\Rightarrow 0<Y<3$

Now, differentiating (i) to get its PDF.

$f_Y(t)=(\dfrac t3)'=\dfrac 13$

Thus, distribution function:

$P(Y\le r)=\{\dfrac r3, 0<r<3 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0,\ \ \ otherwise$

And density function:

$f_Y(t)=\dfrac 13;0<t<3 \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0,\ \ \ otherwise$