Let X1 be uniform over 0 1 and X2 is uniform over 0 3 Suppos

Let X1 be uniform over (0, 1) and X2 is uniform over (0, 3). Suppose they are independent. Find (a) P(1 ? X1 + X2 ? 2); (b) the density of X1 + X2.

Solution

Given density function for X₁ = 1 and density function for X₂ = 1/3,

so X₁+X₂ = 3/5 for 0<x<1

= 1/5 for 1<x<3  

a) So P(1<X<2)= 1/5

b)f_X(x) = 3/5 for 0<x<1

= 1/5 for 1<x<3