91Ó°ÊÓ

Compute the hazard rate function of \(X\) when \(X\) is uniformly distributed over \((0, a)\).

Verify that the gamma density function integrates to 1 .

Let \(X\) be a normal random variable with mean 12 and variance \(4 .\) Find the value of \(c\) such that \(P[X>c\\}=.10\).

Compute the hazard rate function of a gamma random variable with parameters \((t, \lambda)\) and show it is increasing when \(t \geq 1\) and decreasing when \(t \leq 1\).

Let \(X\) be a continuous random variable having cumulative distribution function \(F\). Define the random variable \(Y\) by \(Y=F(X)\). Show that \(Y\) is uniformly distributed over \((0,1)\).

(a) A fire station is to be located along a road of length \(A, A<\infty\). If fires will occur at points uniformly chosen on \((0, A)\), where should the station be located so as to minimize the expected distance from the fire? That is, choose \(a\) so as to minimize \(E[|X-a| \mathrm{I}\) when \(X\) is uniformly distributed over \((0, A)\). (b) Now suppose that the road is of infinite length-stretching from point 0 outward to \(\infty\). If the distance of a fire from point 0 is exponentially distributed with rate \(\lambda\), where should the fire station now be located? That is, we want to minimize \(E[|X-a|]\) where \(X\) is now exponential with rate \(\lambda\).

Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked is an exponential random variable with parameter \(\frac{1}{20}\). Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is the probability that she would get at least 20,000 additional miles out of it? Repeat under the assumption that the lifetime mileage of the car is not exponentially distributed but rather is (in thousands of miles) uniformly distributed over \((0,40)\).

If \(X\) is uniformly distributed over \((0,1)\), find the density function of \(Y=e^{X}\).

Find the distribution of \(R=A \sin \theta\), where \(A\) is a fixed constant and \(\theta\) is uniformly distributed on \((-\pi / 2, \pi / 2)\). Such a random variable \(R\) arises in the theory of ballistics. If a projectile is fired from the origin at an angle \(\alpha\) from the earth with a speed \(v\), then the point \(R\) at which it returns to the earth can be expressed as \(R=\left(v^{2} / g\right) \sin 2 \alpha\), where \(g\) is the gravitational constant, equal to 980 centimeters per second squared.