The time, in 100 hours, that a student uses her game console over a year is a random variable X with probability density function x if 0 < x < 1 f(x) = { 2 – x if 1 < x < 2 O otherwise. The power (in number of kilowatt hours) expended by the student's game console each year is 48X² + 26. For these problems, please ensure your answers are accurate to within 3 decimals. Part a) Find the mean amount of power expended by the student's game console per year. Part b)

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Chapters: Normal and Exponential random variables. Q: Let X be an exponential random variable with rate 1. If a and b are positive numbers, then P(X > a + b|X > b) = P(X > a). a. Explain why this is called the memoryless property b. Show that for an exponential rv X with rate 1, P(X > a) = e-ad.

Use 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).