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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- 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.please quickly thanks !Normal distribution. The moment generating function of a random variable U is M(t) = (1 - 8t)^-7. Find: i)Probability density function, ii)Mean, iii)Variance of U.
- Let X=c with probability 1. Plot the PDF (probability distribution function) of X. Please show all work.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 =How to solve
- 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?The score of a student on a certain exam is going to be a number between 0 and 1. Think of it as the percentage of correct answers on the exam. Suppose the student passes the exam if their score S is at least 0.60 and fails otherwise. If the probability density function of a student score S is given by 4x f(a) = 4 – 4x < <1. 0. otherwise If the student passes the exam, what is the probability that their score S was at least 0.75? O 18.38% O 12.50% O 30.86% O 39.06%
- gn X Three couples and two single individuals have been invited to an investment seminar and have agreed to attend. Suppose the probability that any particular couple or individual arrives late is 0.41 (a couple will travel together in the same vehicle, so either both people will be on time or else both will arrive late). Assume that different couples and individuals are on time or late independently of one another. Let X = the number of people who arrive late for the seminar. (a) Determine the probability mass function of X. [Hint: label the three couples #1, #2, and #3 and the two individuals #4 and #5.] (Round your answers to four decimal places.) P(X = X) X 0 1 2 3 4 5 6 7 8 (b) Obtain the cumulative distribution function of X. (Round your answers to four decimal places.) F(x) X 0 1 2 3 4 5 6 7 Use the cumulative distribution function of X to calculate P(2 ≤ x ≤ 5). (Round your answer to four decimal places.) P(2 ≤ x ≤ 5) = Need Help? Read It Watch It 4Suppose 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.answer only part b .