Suppose a college professor never finishes her lecture before the end of the class period, and always finishes within five minutes after the class period is supposed to end. Let X = time that elapses between the end of the class period and the actual end of the lecture. Suppose the pdf of X is f(x) = kx³/2 0≤x≤ 5 and 0 otherwise. Find the value of k that makes f(x) a legitimate probability density function and use that value of k to find the probability that the lecture ends less than 3 minutes after the class period is supposed to end.

5. Let 0 and a be parameters. Show that the family of functions fo,a(x) = I > 0 (x+0)a+1³ is a probability density function. What can you say about the mean of a random variable that has fo,a as a pdf?

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.

Verify Property 2 of the definition of a probability density function over the given interval. 1 10 f(x)= [4,6]

Suppose a random variable X has a probability density function in img.jpg.  a. Find the value of k that makes this a probability density function. b. Find P(0.4≤X≤1). c. Find P(X≤0.4|X≤0.8) d. Find F(b) = P(X≤b) and sketch the graph of this function.

Suppose a college professor never finishes her lecture before the end of the class period, and always finishes within five minutes after the class period is supposed to end. Let X = time that elapses between the end of the class period and the actual end of the lecture. Suppose the pdf of X is: (image) Find the value of k that makes f(x) a legitimate probability density function, and use that value of k to find the probability that the lecture ends less than 3 minutes after the class period is supposed to end.

Let X be a random variable that follows the beta distribution. This random variable is continuous and is defined over the interval from 0 to 1. The probability density function is given by                                         whereand are integers, whose values determine the shape of the probability density function. Because X varies between 0 and 1, we can think of X as the probability that some event (say) E occurs or the proportion of times an event occurs in some population. For example, E could denote the event that a critical part in a newly designed car will lead to a catastrophic failure in accidents at high speeds.   The expected value (i.e., mean) of this random variable is []. That is,                                             . The Excel commands for the beta random variable are           =beta.dist(x,,,true,0,1) for the cumulative probability distribution, and           =beta.dist(x,,,false,0,1) for the probability density function. (a) Now, think in Bayesian terms.…