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- 5. Let 0 and a be parameters. Show that the family of functions fo,a(x) = I > 0 (x+0)a+1³ is a probability density function. What can you say about the mean of a random variable that has fo,a as a pdf?Based on this density, what is the probability that X is between 0.5 and 1.5? Possible answers are 1/2, 1, 1/3, 3/4. Please show steps on how answer was obtained.According to a study, the maximum temperature T (in degrees Celsius) of each day during the spring has a distribution with the following function density: 21Question 2. A continuous random variable, X has the following probability density function: S(x) = cx² 0sxs10 = 0 otherwise a) Find the value of c. b) Find the mean, median, mode, standard deviation and coefficient of variation. c) Find the probability that X is greater than 3.0 given that X is between 2.0 and 5.0.1. Each rear tire on an experimental airplane is supposed to be lled to a pressure of 40 pounds per square inch (psi). Let Y1 denote the actual air pressure for the right tire and Y2 denote the actual air pressure for the left tire. Suppose that Y1 and Y2 are random variables with the joint density function attached as a photo. a) Find k. b) Find the joint distribution function for Y1 and Y2. c) Find the probability that both tires are under filled.2. Assume that if a certain scale in a laboratory has been used too long without recalibration the probability density function of the measurement error is f(x) = 1 – 0.5x for 0Use R to complete the following. R code and R output are requested for solution. Please keep your solution clear and easy to read. Unclear solution will not be grade. O for manual solution. Carry out a simulation experiment to illustrate the central limit theorem when the population distribution is arcsine with probability density function f(x) = 1 T√x(1-x) x = (0, 1). Consider the four sample sizes n = 1, n = 3, n = 5, n = 10, and in each case use 10,000 replications. Note: To simulate a random sample of size n from the arcsine distribution in R, use the command rbeta(n,1/2,1/2).The density function of the continuous random variable X is shown below. When c≈ 0.163, Find: (1) the mode, (2) the median, and (3) the mean. Jcsin (2 -0.2x) for 1≤ x ≤ 10 f(x) = {0 elsewhere mahamdy@uoanbar.edu.iq4.32 Weekly CPU time used by an accounting firm has probability density function (measured in hours) given by | (3/64)y²(4 – y), 0Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON
A First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON