Answer in Statistics and Probability for yoki #204730

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

Solution:

“Xsim U[0,1]”

“Y=3Xn\Rightarrow X=dfrac Y3”

“P(Yle r)=P[0le Xle 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(Yle r)={dfrac r3, 0<r<3n\ 0, otherwise”

And density function:

“f_Y(t)=dfrac 13;0<t<3n\ 0, otherwise”