Suppose it is known that the proportion of students who get an A in a particular statisticscourse follows a beta distribution with a = 5 and 3 = 2, having probability density function¹(1-x) ³-¹,f(x) =I(a + B)r(a)r(3)"0 < x < 1.(a) Find the probability that more than half of the students will get an A in this class thissemester.(b) On average, what proportion of students are expected to get an A in this class thissemester? If you know the form for the expected value for a Beta random variable, youdo not need to reprove it.

A continuous random variable X is said to follow Beta distribution with parameters α and β if its…

Answered: Suppose it is known that the proportion…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter13: Probability And Calculus

Section13.CR: Chapter 13 Review

Problem 29CR

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2. The time (in minutes) required for technicians to perform a specific task is believed to follow an exponential distribution with probability density function 0.2e-0.2t if t>0 { f(t) = 0ift<0 (a) What is the mean time required by technicians to perform the task and its variance? (b) Calculate the probability that a technician can complete the task in less than 7 mi Post
4. A continuous random Variable X has probability Density function defined by f(x) = 5-5x; 0<=x<=1, =0 otherwise, find the mean and the variance of the distribution.
Suppose that the random variable X has the probability density function c(1- - x2) for - 1< x <1 f(x) = { elsewhere What is the variance of X A 1/3 1/5 c) 1/2 1/4
2B. (a) An exponential distribution has probability density function ƒ(x) = ¹²e-x/a, x ≥0. α Show that the mean of the distribution is a and the variance is a². (b) A machine contains two belt drives, of different lengths. These have times to failure which are exponentially distributed, with mean times a and 2a. The machine will stop if either belt fails, and the failures of the belts are independent. Show that the chance that the machine is still operating after a time a from the start is e 32
A variable X is distributed at random between the values 0 and 4 and its probability density function is given by: f(x) = kx³ (4- x)². Find the value of k, the mean and standard deviation of the distribution.
Suppose a special type of small data processing firm is so specialized that some have difficulty making a profit in their first year of operation. The probability density function that characterizes the proportion Y that make a profit is given by ky² ( 1 - y)3 0
An entomologist is studying a rare type of insect. He finds that the time from birth until death (in years, denoted by T) of a randomly selected member of this insect population has a Gamma distribution with parameters a = 2 and B = 6. Find the average life of a random insect of this type. Find the probability that one of this random insects survives longer than 20 years.
a) A random variable X has an exponential distribution with probability density function given by f(x) = {ledx; for x 2 0 lo ,elsewhere Find the 75th percentile of X. b) A certain brand of light bulb has a lifespan which is exponentially distribution with a mean of 35 days. 1. What is the probability that a randomly selected light bulb of this brand will last more than 45 days? II. If it is known that a light bulb has already lasted more than 25 days, what is the probability that it will at least last for an additional 50 days. (Hint: Memoryless Property) c) Suppose that the amount of time it takes for someone to finish a task, denoted by X, has the uniform distribution between 20 and 80 minutes. 1. Write a suitable probability density function for the time it takes someone to finish the task. II. Using the density function in (1) above derive the mean of X. II. Derive the variance of X using the results obtained in (I) and (I) above. IV. Find the probability that an individual takes…

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