Homework Soursechoose a number U from the interval 01 with uniform distribu
choose a number U from the interval [0,1] with uniform distribution. Find the cumulative distribution and density for the random variables (a) Y= U+2. (b) Y= U^3
choose a number U from the interval [0,1] with uniform distribution. Find the cumulative distribution and density for the random variables (a) Y= U+2. (b) Y= U^3
choose a number U from the interval [0,1] with uniform distribution. Find the cumulative distribution and density for the random variables (a) Y= U+2. (b) Y= U^3
Solution
a) U+2 WILL SHIFT THE LOCATION OF THE DISTRIBUTION.
NOW THE VARIABLE WILL CHANGE TO Uniform [2,3].
So, the cdf is just int( f(t), 2, u) = F(u)=u , 2<u<3
0 for u <2 and 1 for u>3
b)IN THIS WE WILL CHANGE THE DIMENSION
Let V=g(U)=U^3.
Then U=V^1/3
and du/dv=1/3*V^-2/3
LET DISTRIBUTION U =f(u) and LET distribution of V=h(v)
so V is distributed as.
h(v) = f(g^-1(U))*abs(du/dv) = 1*abs(du/dv) = 1/3*V^(-2/3)
The cdf is int( h(t), 0, v) = v^(1/3)
![choose a number U from the interval [0,1] with uniform distribution. Find the cumulative distribution and density for the random variables (a) Y= U+2. (b) Y= U](/WebImages/29/choose-a-number-u-from-the-interval-01-with-uniform-distribu-1078909-1761566282-0.webp "choose a number U from the interval [0,1] with uniform distribution. Find the cumulative distribution and density for the random variables (a) Y= U+2. (b) Y= U")
