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- . Let the waiting time 'w' has a gamma distribution with a = k and b = 12. Write the corresponding probability density function and also obtain the average waiting time.a) Show for the constant c=g, the function f(x) = c(x+x²) for 0 < x < 1 is a proper density of a probability distribution on (0, 1). b) Find the corresponding cumulative distribution function F(x). c) Find the expected value and standard deviation of the random variable X with density5. 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?
- Students arrive at a lecture theatre independently. Suppose the number of students arriving in an hour follows a Poisson distribution with mean 10. Let 7 (in hours) be the time required to wait for 5 students to arrive. Derive the probability density function of T.The probability density function for the continuous random variable X is given by: (А(х? — 2х + 21) 0Let X1,., Xn all be independent exponential distributions all with the same parameter 2. What is the distribution of the minimum of (X1,. a) what is the distribution of the smallest order statistic Y, ? b) Specify both the pdf of the distribution and describe the distribution and its parameters. ,Xn)? In other words:Write CLearly plzQ1. 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 density curve consists of the line segment connecting the points (0,1) and (0.5,1) and the segment connecting(0.5, 1) to the x-axis.a. Determine the coordinate point where the second segment crosses the x-axis.b. Determine the slope of that segmentc. Determine the equation of the line containing this segment (y = mx + b)d. Calculate the probability P(X > 1)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.…