A man bought a car for $8,200 and sold it for 80% of the price two years later. How much did he lose?
2 answers:
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Answer:
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,
