Example. Suppose we choose independently X and Y to be two Exponential(A) random variables. Use their convolution to find the probability density function of their sum Z = X + Y.
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- 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?4. "Time headway" In highway traffic flow the elapsed time between the time that one car finishes passing a fixed point and the instant that the next car begins to pass that point is called time headway. Now, let X = the time headway for two randomly chosen consecutive cars on a highway during a period of heavy traffic flow. The following probability density function of X is suggested by traffic experts: f(x) = 0.15*e-0.15(x-0.5) for (x ≥ 0.5 sec) 0 otherwise a. Draw f(x) from x = 0 to x= 10 sec. b. Find P(X ≤ 5 sec) and show it on the figure you have drawn. c. What is E[X] and Var[X] ? d. Find Cumulative Distribution Function of X.D6) Finance Suppose that the stock price follows geometric Brownian motion dXt = 0.04 Xt dt + 0.2 Xt dWt. Write the probability density function for the distribution of the stock prices in 2 years if the current stock price is 80.
- 1. What two properties must a function f that is a probability density function exhibit?please quickly thanks !Let y be the random variable with the time to hear an owl from your room's open window (in hours). Assume that the probability that you still need to wait to hear the owl after y hours is one of the following: the probability is given by 0. 47e-4y + 0. 52e-5y Find the probability that you need to wait between 2 and 4 hours to hear the owl, compute and display the probability density function graph as well as a histogram by the minute. Compute and display in the graphics the mean, variance, and quartiles of the waiting times. Please pay attention to the various units of time!
- Suppose a college professor never finishes her lecture before the end of the class period, and always finishes within five minutes after the class period is supposed to end. Let X = time that elapses between the end of the class period and the actual end of the lecture. Suppose the pdf of X is: (image) Find the value of k that makes f(x) a legitimate probability density function, and use that value of k to find the probability that the lecture ends less than 3 minutes after the class period is supposed to end.ONLY NEED PART ETwo friends play a ball picking game where one wants to avoid the ‘black’ ball: The two friends each have a bag with 2 balls (1 black and 1 red) and they each randomly choose one ball out of their bag. Whoever picks a black ball must pay the other person $1 no matter what the other friend selected. If both friends choose the red ball, nobody wins or loses any money. Say you played this game with you friend. Find the probability mass function (pmf) for your net winnings (in $) from this game. Display it in table form.
- Directions: The amount of time required to serve a customer at a bank has an exponential density function with mean 4 minutes. 1. Find the probability that a customer is served in less than 3 minutes. Probability = 2. Find the probability that serving a customer will require more than 6 minutes. Probability =A random variable X has probability density function (x² + 2x + 4) for 0 < x < 3 f(x) = otherwise. Which of the following gives the cumulative distribution function F(x)?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?