1. Let X and Y be independent U[0, 1] random variables, so that the point (X,...

1. Let X and Y be independent U[0, 1] random variables, so that the point (X, Y) is uniformly distributed in the unit square.

Let T = X + Y.

(a) Find P( 2Y < X ).

(b). Find the CDF F(t) of T (for all real numbers t).

HINT: For any number t, F(t) = P ( X <= t) is just the area of a part of the unit square.

(c). Find the density f(t).

REMARK: For a number t and a small dt, f(t) dt is the approximate probability that T lands in the interval [t, t+ dt ]. For example, f(0.7) *(0.01) is the approximate probability that T is between 0.7 and 0.71 . Where does (X,Y) have to land for T to be between 0.7 and 0.71 ? Now consider how the probability that T lands in [t, t + 0.01 ] changes as t increases, first from 0 to 1 and then from 1 to 2. Compare with your answer to (c).

Expert Solution

Here we have transformation technique of random variables to solve the problem by using Jacobian.

orchestra answered 2 years ago

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