Suppose that a die is rolled twice. Let X be the minimum value to appear in the two rolls. Construct the table of the probability mass function of X. W Obtain the distribution function of X. Write out the moment generating function. Using the second factorial moment of X, compute the variance.

Assume that the amounts of weight that male college students gain during their freshman year are normally distributed with a mean of u 1.3 parts (a) through (c) below. a. If 1 male college student is randomly selected, find the probability that he gains between 0 kg and 3 kg during freshman year. The probability is (Round to four decimal places as needed.) b. If 16 male college students are randomly selected, find the probability that their mean weight gain during freshman year is between 0 kg and 3 kg. The probability is (Round to four decimal places as needed.) c. Why can the normal distribution be used in part (b), even though the sample size does not exceed 30? OA. Since the weight gain exceeds 30, the distribution of sample means is a normal distribution for any sample size. OB. Since the original population has a normal distribution, the distribution of sample means is a normal distribution for any sample size. and a standard deviation of a 5.1 kg. Complete OC. Since the…

Suppose we roll 3 fair dice and let X be the number of twos we roll.(a) Find the probability mass function of X.(b) Find the expected value of X.(c) Find the variance of X.(d) Find the probability-generating function of X.(e) Find the moment-generating function of X

Airlines sometimes overbook flights. Suppose that for a plane with 50 seats, 55 passengers have tickets. Define the random variable Y as the number of ticketed passengers who actually show up for the flight. The probability mass function of Y appears in the accompanying table. y 45 46 47 48 49 50 51 52 53 54 55 p(v) 0.05 0.10 0.12 0.14 0.25 0.17 0.03 0.06 0.05 0.02 0.01 Calculate V(Y) and oy. (Round your variance to four decimal places and your standard deviation to two decimal places.) V(Y) Determine the probability that Y is within 1 standard deviation of its mean value.

Q2

Let X be a Poisson random variable with rate X. Find the mgf of X.

he chance of an IRS audit for a tax return with over $25,000 in income is about 2% per year. We are interested in the expected number of audits a person with that income has in a 10-year period. Assume each year is independent. Give the distribution of X.X ~

Let the random variable T follow a Student's t distribution with 9 degrees of freedom. State the distribution of the random variable T. Group of answer choices T∼Student′st(9) T∼Student′st(8) T∼Normal(0,1) T∼Student′st(0,1)     Compute the probability that T is between −0.1 and 0.6.    True or False:  The larger the degrees of freedom in a Student's t distribution, the closer to a standard normal distribution the Student's t distribution becomes.