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- 3) Find the mean and variance of the random variable X with probability function or density f(x). (a)f(x)={k(1-x), –15xs1 O otherwise (b) ƒ(x) = k/3× , x = 1,2,3,... Note: in this problem, you need to first find the values of k.Consider the continuous random variable with probability density function fx(x) = 3x² for 0Example 3: The probability density ƒ(x) of a continuous random variable is given by f(x) 3 ce , -ma) Let X be a continuous random variable with the following probability density function (2x3 f(x)={11 2 +5) , 0Let X be the proportion of new restaurants in a given year that make a profit during their first year of operation, and suppose that the density function for X is ƒ(x) = 20x³(1 − x) Find the expected value and variance for this random variable. E(X) = = Var(X) 0 ≤ x ≤ 11. A continuous random variable X is defined by (3+x) f(x) - 3sxs-1 %3D 16 (6-2r) -1sxs1 16 (3-x) -1sxs3 16 a. Verify that f(x) is density. Explain b. Find the mean 2. If the probability density function of random variable is given by f(x) = wch 1srs2 sech x a. Find the mean b. Find the total area 3. If the probability density function of random variable is given by f(x) = sinh 2x 1sxs2 a. Find the mean b. Find the total area c. Illustrate the graph (use bell shaped figure) II IIQ6. A continuous random variable has probability density function = { / (2x - 3) ,15x52 0 , otherwise f(x) = The find the value of P(1.5 ≤ x ≤ 4)A continous random variable has density function f(x)=(3x?)/(63) for Osxse, for some 0>0, and zero otherwise. What is the ratio of the mode to the median for this distribution? A. 2(1/3) B. 0/(21/3) O C.e D. 2(-1/3)Suppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.Recommended textbooks for youMATLAB: An Introduction with ApplicationsStatisticsISBN:9781119256830Author:Amos GilatPublisher:John Wiley & Sons Inc
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