The variance of a discrete random variable X with probability density function P, denoted by Var[X], is Var[X] = E,p - iE[X]? Select one: True False
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- Let X be a random variable with density function f(x) = { k(1-x²), if 0≤x≤ 1; otherwise. Find k, together with the expectation and the variance of the random variable Y defined as Y = 3X - 1. fandq11The table below shows an incomplete probability density function for X , the number of imperfections per batch of jeans that just came off a production line. What should the missing probability be? Provide the final answer as a decimal. x P(X = x) 0 0.88 1 0 2
- For the probability density function f defined on the random variable x, find (a) the mean of x, (b) the standard deviation of x, and (c) the probability that the random variable x is within one standard deviation of the mean. 1 f(x) = x, [2,4] ..... a) Find the mean. (Round to three decimal places as needed.) b) Find the standard deviation. O = (Round to three decimal places as needed.) c) Find the probability that the random variable x is within one standard deviation of the mean. The probability is (Round to three decimal places as needed.)Let X be a continuous random variables with with the following probability density function.Answer the following questions.Find the value of α such that f(x) is a probability density function. Make sure thatyou show all the steps.Find the cumulative distribution function of X. Show it on a graph. Make sure thatyou show all the steps.Find the expected value and variance of X. Make sure that you show all the steps.Find the probability density function of Y = X2. Make sure that you show all thesteps.2. Consider a gas station's daily sales. Let Y be the volume of gas sold per day in 1000s of gallons. Assume the distribution of Y follows this pdf: 1sxs 2 S (v) . else What value of k is required for the above function to be a valid pdf? b. Verify that the function given is a valid probability density а. function c. Determine the cumulative distribution function (cdf) of the random variable d. Compute the probability that there is exactly 1500 gallons of sales on a given day (Hint: Find?(r - 1.5)) = 1 CO0
- The lifetime of a machine part has a continuous distribution: Osx<40 fylx) = (10+ x)? else a. Calculate the probability that the lifetime of the machine part is more than 6. b. Calculate the expectation of life of this machine C. Calculate the variance Calculate the probability that the lifetime of the machine part is more than 4 given that is lasts for less than 12 d.For the probability density function f defined on the random variable x, find (a) the mean of x, (b) the standard deviation of x, and (c) the probability that the random variable x is within one standard deviation of the mean. 1 f(x) = x, [5,9] 28 a) Find the mean. %3D (Round to three decimal places as needed.) b) Find the standard deviation. (Round to three decimal places as needed.) c) Find the probability that the random variable x is within one standard deviation of the mean. The probability is. (Round to three decimal places as needed.)Let X be a continuous random variable with probability density function xe-X x20 fxlx) = 0 otherwise The value of a Select one: a. can be any positive real number b. must be 1 c. can be any real number d. can be any positive integerLet x be a continuous random variable that has a normal distribution with u = 50 and o = 8. Find the probability P(30 ≤ x ≤ 39).