Let X equal the number of hours that a randomly selected person from City A reads the news online each day. Suppose that X has a continuous uniform distribution on the interval [1, 4]. Let Y equal the number of hours that a randomly selected person from City B reads the news online each day. Suppose that Y has a continuous uniform distribution on the interval [1,5]. Assume that X and Y are independent. Determine P(Y> X).

Step-by-step explanation:

Recall that since X is uniformly distributed over the set [1,4] we have that the pdf of X is given by if and 0 otherwise. In the same manner, the pdf of Y is given by if and 0 otherwise.

Note that if Y is in the interval (4,5] then Y>X by default. So, in this case we have that P(Y>X| y in (4,5]) = 1. We want to calculate the probability of having Y in that interval . That is

. Thus, .

We want to proceed as follows. Using the total probability theorem, given two events A, B we have that

In this case, A is the event that Y>X and B is the event that Y is in the interval (4,5].

If we assume that X and Y are independent, then we have that the joint pdf of X,Y is given by when . We can draw the region were Y>X and the function h(x,y) is different from 0. (The drawing is attached). This region is described as follows: and , then (the specifics of the calculations of the integrals are ommitted)

Thus,

a - base

b - exponent

so,