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:
"X\\sim U[0,1]"
"Y=3X\n\\\\\\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\n\\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0,\\ \\ \\ otherwise"
And density function:
"f_Y(t)=\\dfrac 13;0<t<3\n\\\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ 0,\\ \\ \\ otherwise"
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