For the following probability density function f(x) = (cx²; 0

a. Verify that h(x) = = 8 3x³ 0 each 1≤ x ≤2 otherwise is a valid probability density function.

Consider the probability density function 1 if x -4 + (x+4) Find the median. Select one: -3 О -5 O 1 O 0.5

A continuous random variable, X, has the following probability density function. k cos for 0sxs1 f(x)=- for all other values of x (a) Show that k = Find p( x sxs). (b) (c) Find the median of X.

Which of the following tables shows a valid probability density function? Select all correct answers.   Select all that apply:   x P(X=x) 0 0.6 1 0.01 2 0.14     x P(X=x) 0 3/10 1 1/10 2 2/5     x P(X=x) 0 1/8 1 1/4 2 5/8     x P(X=x) 0 0.13 1 0.09 2 0.45 3 0.27 4 0.06     x P(X=x) 0 −1/5 1 3/10 2 1/2 3 3/10 4 1/10     x P(X=x) 0 1/2 1 1/8 2 3/8

(1) Explain why the following functions are, or are not, probability density functions: (a) f(x) = ² for x = [1, √e] (b) f(x) = 5 (9x² – x4) for x = [0,3]

Let f(x) = k(4x –x²) if 03. (a) For what value of k is f(x) a probability density function? (b) For that value of k, find P (x>2). (c) Find the mean.

Let X denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. Suppose that X has the following probability density function (called the Rayleigh probability density function). f(x) S (x/0²) e-x²/(20²) x > 0 {o otherwise (a) If 0 = 90, find the probability that the vibratory stress is between 73 and 190. (b) If = 90, then 83% of the time the vibratory stress is greater than what value?

Let X denote the vibratory stress (psi) on a wind turbine blade at a particular wind speed in a wind tunnel. Suppose that X has the following probability density function (called the Rayleigh probability density function). ♫ (x/0²) e-x²/(20²) x > 0 f(x) = {(x otherwise (a) If 0 = 110, find the probability that the vibratory stress is between 99 and 331. (b) If 0 = 110, then 79% of the time the vibratory stress is greater than what value?

A random variable X has probability density function (x² + 2x + 4) for 0 < x < 3 f(x) = otherwise. Which of the following gives the cumulative distribution function F(x)?