Thread: Uniform density and integral

Let X_1 and X_2 be independent random variables having the uniform density with a = 0 (alpha = 0) and b = 1 (beta = 1). Find expression for the distribution function of Y = X_1 + X_2 for 1 <y < 2. According to the book, the final answer is 1 - 1/2(2 - y)^2, but I cannot construct the integral expression, from which I would get this final result. Please help me construct this integral. Can you please show me how to calculate the area of appropriate region of the unit square?

Originally Posted by bluesilver

Let X_1 and X_2 be independent random variables having the uniform density with a = 0 (alpha = 0) and b = 1 (beta = 1). Find expression for the distribution function of Y = X_1 + X_2 for 1 <y < 2. According to the book, the final answer is 1 - 1/2(2 - y)^2, but I cannot construct the integral expression, from which I would get this final result. Please help me construct this integral. Can you please show me how to calculate the area of appropriate region of the unit square?

As far as I can see there is something wrong with this question. If X1 and X2
are independent RV ~U(0,1) (which appears to be what the question is saying,
then the distribution of Y=X1+X2 is the symetric triangular distribution on (0,2).

I can see no way of reconciling this with

p(y) = 1 - 1/2(2 - y)^2

wherever I put brackets in this expression.

The attachment shows the histogram of 100000 realisations of y, and this
is clearly the triangular distribution.

RonL