Let X be a continuous random variable with a density function: |x| f(x) = - 2
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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. fandLet x be a continuous random variable with density function +K 9. 0The weekly sales for a drinking water product (in 1000s of liters) is a continuous random variable Y with probability density function (pdf)f(y) v-1) 0Prove that for a > 0, the function 1 f(2) = -z(a/2)–1e-z/2 2ª/2T°(a/2) defines a probability density function on (0, x). This distribution is known as the x² with degrees of freedom being any arbitrary positive real number.Let X be a RV with the following probability density function f (æ) = { (6+1) xº, 0Let X and Y be two continuous random variables with joint probability density function given by: (12x(1– x). %3D 0, otherwise Then the cov(X,Y)is: None of these -1/36 1/50Given that the probability density function of the continuous random variable X is given as follows, what is E (6X + 3X²) equal to? f(x)={ 2(1-x), 0<x<1 0, e.w a)15/18 b)5/2 c)1/18 d)1/3Let X be a random variable with the following probability density function: f (x) = (1 – a2), –1 < x < 1 4 and f (x) = 0 otherwise. Give the integrals for the following: a) The mean b) The variance c) P[ -0.75 < X < 0.5 ]. d) the CDFFind the value of k that makes the given function a probability density function on the specified interval. f(x) = kx 2 ≤ x ≤ 6 k= || =The total number of hours, measured in units of 100 hours, that a family runs a vacuum cleaner over a period of one year is a continuous random variable X that has the density function shown below. Find the probability that over a period of one year, a family runs their vacuum cleaner (a) less than 110 hours; (b) between 55 and 80 hours. f(x) = X, 0a) What is the value of k?Recommended 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