Question: An automobile insurance plocy will pay for the damage to both the poli

Question

Question: An automobile insurance plocy will pay for the damage to both the policyholder's car and the other driver's car in the event that the policyholder is responsible for an accident. The payment for the damage to the policyholder's car, X, has a marginal density function of 1 for 0<x<1. Given that X=x, the payment for the damage to the other person's car, Y, has a conditional density of 1 for x<y<x+1. If the policyholder is responsible for the accident, what is the probability that the payment for the damage to the other driver's car will exceed 0.5?

Explanation / Answer

The probability to compute is P(Y 0.5).
We are given that X is
uniform on [0, 1] and that, given X = x,
Y is uniform on [x, x + 1].
Thus, fX(x) = 1 for 0 x 1 and h(y|x) = 1 for x y x + 1, 0 x 1.
Hence
f(x, y) = fX(x)h(y|x) = 1, 0 x 1, x y x + 1.
In particular, this means that the joint density is uniform on the region (a parallelogram—
sketch!) 0 x 1, x y x + 1.
Hence probabilities can be computed as ratios of areas.
The entire parallelogram has area 1, and the part of this parallelogram
on which y 0.5 has area 1 (1/2)0.5^2=7/8 by elementary trigonometry.
Hence
P(Y 0.5) = (7/8)/1 = 7/8.

Related Questions

Navigate

• Browse (All)
• Browse Q
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.