1. Consider an infinite sequence of independent experiments, where in each experiment we take k balls, labeled 1 to k, and randomly place them into k slots, also labeled 1 to k, so that there is exactly one ball in each slot. For the nth experiment, let Xn be the number of balls whose label matches the slot label of the slot into which it is placed. So X1, X2, . . . is an infinite sequence of independent and identically distributed random variables.

   1. (a) Find the expected value and variance of Xn.

   2. (b) Use the central limit theorem to approximate the probability that in the first 40 experiments

      the total number of balls whose label matches their slot label is greater than 40 (this means find

      an approximation to P (?40 Xn > 40) using the central limit theorem.)n=1

The number of arrivals per minute at a bank located in the central business district of a large city was recorded over a period of 200​ minutes, with the results shown in the table below. Complete​ (a) through​ (c) to the right.

a. Compute the expected number of arrivals per minute.

μ=

​(Type an integer or decimal rounded to three decimal places as​ needed.)

b. Compute the standard deviation.

σ=

​(Type an integer or decimal rounded to three decimal places as​ needed.)

c. What is the probability that there will be fewer than 2 arrivals in a given​ minute?

P(x<2)=

​(Type an integer or decimal rounded to three decimal places as​ needed.)

1A. Map out the correct chromosome map for four genes given the following information

X-Z = 18%X-G = 10%Z-Y=27%Z-G=28%X-Y=9%

1B. Hemophilia is a type of blood clotting disorder that is an example of a sex-linked (X-linked) recessively inherited trait. Polydactyly is an example of a dominantly inherited disorder, where individuals with the dominant allele have extra fingers or toes. Josh has the normal number of fingers and toes and has hemophilia. Josh is married to Olive. Olive does not have hemophilia but has 12 fingers and 14 toes. Olive’s father had hemophilia but has a normal number of fingers and toes.

What is the probability of Josh and Olive having a child with hemophilia and polydactyly?

. A certain supermarket has both an express checkout line and a super-express checkout line. Let X1 denote the number of customers in line at the express checkout at a particular time of day, and let X2 denote the number of customers in line at the super-express checkout at the same time. Suppose the joint probability distribution of X1 and X2 is as follows

x2

0 1 2

0 0.1 0. 1 0.0

x1 1 0.1 0.2 0.1

2 0.0 0.1 0.3

(a) Find P(X1 = X2). (4 pts.)

(b) Find P(X1 6= 0). (4 pts.)

(c) Find the marginal distribution of X1. (4 pts.)

(d) Find Cov(X1, X2). (4 pts.)

(e) Are X1 and X2 independent? Why? (4 pts.)

Sarah’s Muffler Shop has one standard muffler that fits a large variety of cars. Sarah wishes to establish a reorder point system to manage inventory of this standard muffler.

Use the following information to answer the questions.

You must show your calculations—do not just give a number without showing how you arrived at the number.

1. Determine the best order size
2. What is the reorder point?

Customers arrive at a cafe every 2 minutes on average according to a Poisson process. There are 2 employees working at the bar providing customer service, i.e., one handling customer orders and another handling payments. It takes an average of 1 minute to complete each order (exponentially distributed). Based on the above:

f. What are the service time probability density and cumulative distribution functions?

g. What percentage of customer orders will be prepared in exactly 2 minutes?

h. What are the chances it will take between 3 and 4 minutes to prepare a customer’s order?

i. What is the average service rate for completing orders?

j. What is the average number of customers waiting to order?

k. What is the average number of customers at the cafe?

l. On average, how long does it take to serve a customer?

1) 4 ballpoint pens are selected without replacement at random from a box that contains 2 blue pens, 3 red pens, and 5 green pens.
If X is the number of blue pens selected and Y is the number of red pens selected

a. Write the “joint probability distribution” of x and y.

b. Find P[(X, Y ) ∈ A], where A is the region
{(x, y)|x + y ≤ 2}.

c. Show that the column and row totals of a Table and give the marginal distribution of X alone and of Y alone

d. Find the conditional distribution of Y, given that X = 1; namely, f (Y|1)=?

e. Use it to determine P(Y = 0 | X = 1).

f. Show that if the random variables X and Y are statistically independent or NOT independent.

A. A new shopping mall is considering setting up an information desk manned by one employee. Based on information obtained from similar information desks, it is believed that people will arrive at the desk at the rate of 20 per hour. It takes an average of 2 minutes to answer a question. It is assumed that arrivals are Poisson and answer times are exponentially distributed. i. Find the probability that the employee is idle. ii. Find the proportion of the time that the employee is busy. iii. Find the average number of people receiving and waiting to receive information. iv. Find the average number of people waiting in line to get information. v. Find the average time a person seeking information spends at the desk. vi. Find the expected time a person spends just waiting in line to have a question answered

Let X be the number of packages being mailed by a randomly selected customer at a certain shipping facility. Suppose the distribution of X is as follows: x 1 2 3 4 p(x) .4 .3 .2 .1

a. Consider a random sample of size n = 2 (two customers), and let X be the sample mean number of packages shipped. Obtain the probability distribution of X .

b. Refer to part (a) and calculate P (Xbar is less than or equal to 2.5)

c. Again consider a random sample of size n = 2, but now focus on the statistic R = the sample range (difference between the largest and smallest values in the sample). Obtain the distribution of R.

d. If a random sample of size n = 4 is selected, what is p (Xbar less than or equal to 1.5)?