where am i going wrong - sum of two uniform variates.

Problem : Let

X have a uniform distribution on the interval (1; 3).What is the proba-

bility that the sum of 2 independent observations of X is greater than 5?

Solutions :

Let X1 and X2 be the two independent obeservations.

We have to show P(X1+X2 > 5)=1- P(X1+X2 <=5).

Here f(x1,x2)=1/2 for 1<x1,x2<3

= 0 otherwise

(Is the joint pmf and its range correct??? )

here x1 varies from 1 to 3 and x2 varies from 1 to 5-x1.

Now P(x1+x2<=5)= integral (1<x1<3) integral (1<x2< 5-x1) of 1/2 dxdy

on solving this i am getting answer 2....which is wrong( as probablity can not be more than 1)

the right answer of P(X1+X2 >5)=1/8...

Where i am going wrong please let me know...

thanks....