STAT4101 Exam 2 – Practice Problems November 10, 20061. What is the distribution of X (or (X , Y )) in each of the following situations? If the distributionhas a name, please state it, and if it has parameters, please state those also. If the distribution hasa shorthand notation, you may use that, for example, X ∼ Bin(5, 0.4). (In some situations, thedistribution may not be exact. In that cas e, choose the distribution that best fits the situation.)a) A machine produces parts one at at a time, and each part is judged to be either good ordefective. The probability of a defective part is 0.1. The operator needs to produce 100good parts to fill his quota for the day. Let X be the total number of parts he makes inorder to fill his quota.b) Same machine and operator as above. The boss stops by to watch, and stays while 10 partsare made. Let X be the number of bad parts the boss sees made.c) Flaws occur in a rope at random, but on average are five feet apart. Let X be the numberof flaws in a fifty foot piece of rope.d) A box of 100 nails has 5 that are defective. A carpenter takes 10 out of the box at randomto make his next project. Let X be the number of defective nails that he took.e) Customers come to a certain post office either to send a package or pick up their mail (butnever both). On average, someone comes in to mail a package every ten minutes, and topick up their mail every five minutes. Let X be the total number of customers that enterthe post offi ce between noon and 1pm.f) A person drops a ball into a box that is one foot long and one foot wide. Assume the ballis equally likely to be anywhere in the box, and let (X, Y ) denote the position of the ballin the box.g) John says meteors hit the earth on average every 100 years. Assuming John is right, let Xbe the number of years until the next meteor hits earth.h) 49 Democrats, 49 Republicans, and 2 Independents are in the Senate. We choose a senatorat random. Let X be the party of this senator.2. Let X ∼ Bin(10, 0.4). Find P (X = 2). Find P (X ≤ 2). Find E(X). (Copies of p. 648 and p.650 will be included in the exam if a problem like this is included.)3. Let X ∼ Geo(p) and Y ∼ NegBin(3, p), where X and Y are independent. Find P (X + Y = 0).Find the m ean and variance of X + Y .4. Suppose that some event occurs according to a Poisson process, with 2 events per minute onaverage. What’s the average length of time until the next event? What’s the probability of noevents in the next minute?5. Suppose calls come into a service center following a Poisson process, average 10 calls an hour. LetX be the number of calls an operator answers in a given hour, so X ∼ Poi(10). With probability0.1, the operator cannot answer the question and must forward it to the supervisor. What’s theexpected number of calls an operator must forward to the supervisor in a given hour? Use thefollowing s teps to help you.a) Assume we already know how many total calls were received (X), let Y be the number ofcalls that were forwarded. What’s the distribution of Y , given X known?b) What’s the e xpectation of Y given X known?c) Now find E(Y ). Remember E(Y ) = E(E(Y |X)).1/2STAT4101 Exam 2 – Practice Problems November 10, 20066. Let the cumulative distribution function of X beF (x) =0 for x < 0x2for 0 ≤ x < 1/21/2 for 1/2 ≤ x < 1x/2 for 1 ≤ x < 21 for x > 2,as in the following plot.F (x)-0.5 0.0 0.5 1.0 1.5 2.0 2.50.00 0.25 0.50 0.75 1.00xa) Find P (X < 1/4).b) Find P (X = 1/2).c) Find P (X > 1|X > 1/2).d) Let Y = 2X. What’s the cumulative distribution function of Y ?7. The probability density function for X is f(x) = c(1 − x2) for −1 < x < 1 and 0 otherwise.a) Show c = 3/4.b) Find the mean and median of X.c) Find P (X = 1/2).d) Find F and use it to find P (0 < X < 1/2).e) Find the density of 2X.8. Let X ∼ Unif(0, 1) and Y ∼ Exp(1), independently.a) What’s the joint density of X and Y ? Remember to state the support.b) Find E(XY ).c) Find P (Y < X).9. Let the joint density of X and Y be uniform over the unit circle.Are X and Y independent? Why or why not?10. P (X = 0) = 1/2, P (X = 1) = 1/4, and P (X = 2) = 1/4. Use the pgf to find