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- A continuous random variable X has probability distribution function f(x) , wheref(x) = k(1 − x2) for − 1 < ? < 1. Calculate the P(X < 0)?Suppose that a random variable X has the following probability density function. SC(36-x²) 0 ≤ x ≤ 6 otherwise f(x) = {C (² Find the expected value of X. (You will need to find the value of the constant C so that f is a pdf.)2. where The random variable, X, has a probability density function (pdf), f(x), and zero otherwise. (a.) (b) x+1, f(x) = {1x, 1 81 -1/2 < x < 0 0Suppose that a random variable X has the following probability density function. SC (64-x²) 0 ≤ x ≤ 8 otherwise f(x) = {C Find the expected value of X. (You will need to find the value of the constant C so that f is a pdf.)If 2 > 0 and X is a continuous random variable with p. d.f. 22xe-dx ,x 2 0 ,otherwise f(x) = Then var(X) is:Q6. 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)Q1. Suppose X is a continuous random variable. Find an example of a probability density function for X giving expected value E(X) = 1 and variance V (X) = 3 if X has . . . (a.) a uniform distribution. (b.) an exponential distribution. (c.) a normal distribution. In each case, if there is no such probability density function, explain why this is so.a) Suppose that X is a random variable having the following probability density function given by (2(0-x) 02 f(x) = -, x > 0 (0, elsewhere Find the value of c such that an interval from x to cx is a (1- a)100% confidence interval for the parameter 0.The probability that none of the three bulbs in a set of traffic lights will have to be replaced during the first 1200 hours of operation if the lifetime X of a bulb (in thousands of hours) is a random variable with probability density function f(x) = 6[0.25-(x-1.5)2] when 1 ≤ x ≤ 2 and f(x) = 0 otherwise is6. Suppose that the random variables X and Y have joint probability density function given by x+y, 09. Let X be a random variable with probability density function, What is Var(X)? wwwwww f(x) = cx(5-x) for 0 ≤ x ≤ 5 otherwise 0 250 21. Consider a random variable X with probability density function given as: 0 if x < 0, 1 if 0 ≤ x ≤ 1, f(x) 2 1 if 1 < <∞. -2x² Verify if f(x) is indeed a density function. If it is, find the distribution function of X.SEE MORE QUESTIONSRecommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON