Probability

A fire station is to be located along a road of length A, A < ∞. 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... (See attachment for full question)

Probability

34. Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked as an exponential random variable with parameter 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 2 ...continues

Probability density function

#39. If X is an exponential random variable with parameter λ = 1, compute the probability density function of the random variable Y defined by Y = log X. #40. If X is uniformly distributed over (0,1), find the density function of Y = e^x.

Give the joint probability mass function

#2. Suppose that 3 balls are chosen without replacement from an urn consisting of 5white and 8 red balls. Let Xi equal 1 if the ith ball selected is white, and let it equal 0 otherwise. Give the joint probability mass function of (a) X1, X2; (b) X1, X2, X3. #6. A bin ...continues

Find the conditional density

(a) Find the conditional density of X, given Y = y, and that of Y, given X = x. (b) Find the density function of Z = XY. (see attachment)

A prisoner is trapped in a cell containing 3 doors

#53. A prisoner is trapped in a cell containing 3 doors. The first door leads to a tunnel that returns him to his cell after 2 days travel. The second leads to a tunnel that returns him to his cell after 4 days travel. The third door leads to freedom after 1 day of travel. If it is assumed that the prisoner will always select do ...continues

The number of winter storms in a good year is Poisson random variable

#65. The number of winter storms in a good year is Poisson random variable with mean 3, whereas the number in a bad year is a Poisson random variable with mean 5. If next year will be a good year with probability 0.4 or a bad year with probability 0.6, find the expected value and variance of the number of storms that will occur. ...continues

Find the joint mass function of X and Y

#39. Choose a number X at random from the set of numbers {1, 2, 3, 4, 5}. Now choose a number at random from the subset no larger than X, that is, from {1,…, X}. Call this second number Y. (a) Find the joint mass function of X and Y. (b) Find the conditional mass function of X given that Y = i. Do it for i = 1, 2, 3, 4, 5. (c ...continues

Two dice are rolled

#40. Two dice are rolled. Let X and Y denote, respectively, the largest and smallest values obtained. Compute the conditional mass function of Y given X = i, i = 1, 2, 3, 4, 5, 6. Are X and Y independent? Why? (Question is also included in attachment)

An urn has m black balls

#14. An urn has m black balls. At each stage a black ball is removed and a new ball, that is black with probability p and white with probability 1 - p, is put in its place. Find the expected number of stages needed until there are no more black balls in the urn. (Question is also included in attachment)