5. Suppose X and Y are two independent Uniform(0, 1) random variables. Use the cumulative distribution function method to find the probability density function of their sum U = X+Y.
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- Assume that the probability that an airplane engine will fail during a torture test is 12and that the aircraft in question has 4 engines. Construct a sample space for the torture test. Use S for survive and F for fail.2. Suppose that the random variable X possess a uniform distribution on the interval (0, 1). Let U=X. Find the marginal probability density function of U using the distribution function method.Battery life between charges for the Motorolla Droid Razr Maxx is 20 hours when the primary use is talk time. The battery life drops to 7 hours when the phone is primarily used for Internel applications over cellular. Assume that the battery life in both cases follows an exponential distribution. a. Show the probability density function for battery life for the Droid Razr Maxx phone when its primary use is talk time. b. What is the probability that the battery charge for a randomly selected Droid Razr maxx phone will last no more than 15 hours when its primary use is talk time? c. What is the probability that the battery charge for a randomly selected Droid Razr Maxx phone will last more than 20 hours when its primary use is talk time? d. What is the probability that the battery charge for a randomly selected Droid Razr Maxx phone will last no more than 5 hours when its primary use is Internet Applications?
- Suppose X is a random variable taking values in the interval [0,2] with probability density function f(x) = 1-x/2. What is the variance of X?1. Two balls are drawn in succession without replacement from an urn containing 4 red balls and 3 black balls. Consider the random variable Y, represents the number of red balls. Find the probability density function of Y.The probability density function of a continuous random variable is given by 1 1 4 f (x) = - 4 for x > 0. a) Find P(X> 5). b) Find the mean of X. c) Find the CDF (cumulative distribution function). d) Use the CDF to find the median of X.
- 6. Suppose X is a continuous random variable with a strictly positive probability density function f(r); and let F(r) be the corresponding cumulative distribution function. Let U be a random variable uniform on the interval (0, 1). Show that Y = F-1(U) has the same distribution as X. Mathematical Statistics Midterm Part One Page 3 of 418.) As a measure of intelligence, mice are timed when going through a maze to reach a reward of food. The time (in seconds) required for any mouse is a random variable Y with a probability density function given by fy(y) = y² -& y ≥ b elsewhere where b is the minimum possible time needed to traverse the maze. What is the cumulative distribution function for Y?For a certain psychiatric clinic suppose that the random variable X represents the total time (in minutes) that a typical patient spends in this clinic during a typical visit (where this total time is the sum of the waiting time and the treatment time), and that the random variable Y represents the waiting time (in minutes) that a typical patient spends in the waiting room before starting treatment with a psychiatrist. Further, suppose that X and Y can be assumed to follow the bivariate density function fXY(x,y)=λ2e−λx, 0<y<x, where λ > 0 is a known parameter value. (a) Find the marginal density fX(x) for the total amount of time spent at the clinic. (b) Find the conditional density for waiting time, given the total time. (c) Find P (Y > 20 | X = x), the probability a patient waits more than 20 minutes if their total clinic visit is x minutes. (Hint: you will need to consider two cases, if x < 20 and if x ≥ 20.)
