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 Class - Undergraduate 400 Level

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 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).

Probability Class - Undergraduate 400 Level

#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 of 5 transistors is known to contain 2 that are defective. The transistors are to be tested, one at a time, until the defective ones are identified. Denote by N1 the number of tests made until the first defective is identified and by N2 the number of additional tests until the second defective is identified; find the joint probability mass function of N1 and N2. ... click for more

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)