Find the probability at least one of the children will

Question

Question 1) Cindy and Jack want to stay up on Christmas Eve to see if Santa Claus is real. Typically their bedtime is 9pm, and their ability to stay up past that is T_c ~ Exp(04) hours for Cindy and T_J~Exp(08) for Jack.

You can assume T_c and T_J are independent.

a) Find the probability at least one of the children will still be awake if Santa Claus arrives at midnioht.

b) An enterprising 8-year-old, Cindy figures out they can increase their chances of seeing Santa Claus if Jack takes a nap while she waits, and they switch turns when she gets too sleepy. What is the expected value and the variance of their combined waiting time under Cindy's plan? (When Jack wakes up, he stays awake for T_J~Exp(08) hours).

Jack rejects Cindy's plan because he doesn't want to miss out. At 9pm, both children set up in front of the chimney and wait.

c) At midnight, mom and dad hear movement in the living room, indicating that at least one child is still awake. What is the probability Jack is still awake?

d) If Jack is the only child still awake at midnight, what is the expected time until both of them are asleep?

Question 2) A computer network works by sending a stream of packets from a server to a client computer. Suppose the size of the stream has a Poisson distribution with mean λ = 1000 packets. Sometimes packets are lost in transition. Assume each packet may be independently dropped with probability p = 10^-3. (This question is slightly different from the version on the practice exam, please read carefully).

a) What is the probability the client receives an entire stream without any dropped packets on the first pass?

b) The client is able to tell whether an incoming stream is incomplete, and if so, it asks the
server to resend the missing packets. When a packet is resent it is called a retransmission and a missing packet is repeatedly retransmitted until it is received at the client (this is a networking protocol called TCP). For a stream with a fixed length of n packets, what is the expected value and variance of the number of packet retransmissions that will occur before the fill] stream is received at the client? (If the full stream arrives in the first pass, number of retransmissions = 0).

c) For a randomly selected stream, what is the expected value and variance of the number of packet retransmissions that will occur before the full stream is received at the client?

Question 3) A and B are two contractors independently bidding on a job. The contract will go to the lowest bidder. Contractor A submits a random bid X~Uniform(0,1) while contractor B submits a random bid Y with a probability density f_y(y) = 2y for 0 < y < 1 and f_y(y) = 0 otherwise.

a) What is the joint density of Xand Y?
b) Find the probability contractor A is awarded the iob.
c) Find the expected value of the winning bid.

Question 4) a) Consider two independent random variables X and Wwith identical variance. Let Y and Z be defined as

Y = aX + bW
Z = cX + dW

where a, b, c, and d are constants. Choose constants a, b, c, and d such that COV(X,Y) < 0, COV(X,Z) < 0, and COV(Y,Z) < 0. If this is not possible, provide a brief explanation why.

b) Let Z = ρX + √(1 + ρ²W), with 1 ≤ p ≤ 1. Find the correlation coefficient between X and Z.

Question 5) The age of a tree can be determined by counting the number of rings on its trunk. One ring is added onto a tree's trunk each year. Moreover, the width of each ring is random and depends on the amount of rainfall in that year. For example, the radius of a four year-old tree is R = X₁ + X₂ + X₃ + X₄ where {X_i} is a sequence of i.i.d. ring widths. Years can be classified as Wet, Dry, or Normal, each occurring with equal probability. The table below summarizes the conditional expected ring width (in inches) and its respective conditional standard deviation.

Annual Ring Width statistics	Wet	Dry	Normal, 0.33333
Probability	0.33333	0.33333
Expected Ring Width in Inches	3	1	2
Standard Deviation of Ring Width, in inches	1.5	0.5	1

You may invoke the Central Limit Theorem and the 68-95-99.7 rule.

a) Compute the expected value and variance of the ring width in a single year.

b) Compute the expected tree radius and its variance of a 50 year old tree.

c) Construct a 95% confidence interval for the radius of a 50-year tree. That is, find a range (a, b) such that there is a 95% probability that the radius of a 50-year old tree will be inside this range. Your interval should be centered on the expected radius of the 50 year old tree.

d) If f_R(r) and F_R(r) are the PDF and CDF for the radius, derive the distribution of the area of the cross section of a 50-year old tree (assuming that the cross section is a perfect circle). (Hint: you may give an answer in terms of either the PDF or the CDF of R. The area of a circle with radius r is Πr². Note that r ≥ 0 and Πr² is monotonic over this range).

Question 6) You are the manager of a bar that sells beer by the pint (16 oz.) and by the pitcher (64 oz.). The actual amount of beer filled in a pint or pitcher is a random variable, denoted X and Y respectively. You can assume these random variables are independent (the actual amount in the first pint/pitcher is independent from the amount in the next pint/pitcher). The means and standard deviations of X and Y are provided in the table below.

It's Friday and you suspect your bartenders may be habitually giving drinks to their friends for free, which is strictly against the bar's policies. Let M and N be random variables representing the number of pints and pitchers sold, respectively. The means and standard deviations of M and N are provided in the table below. Assume M and N are also independent.

		Statistics of X and Y		Statistics of M and N
	Size (ounces)	Mean	Standard Dev	Mean	Standard Dev
Pint	16	14.5	1.3	220	40
Pitcher	64	60.5	3	50	10

a) Find the expected value and variance of the ounces of beer sold in pits on Friday.

b) Find the expected value and variance of the ounces of beer sold in pitchers on Friday.

c) Find the expected value and variance of the total ounces of beer sold on Friday.

d) Let K be the number of kegs used on Friday. A standard keg holds 15.5 gallons (approximately 1980 oz.). Find the expected value and standard deviation of K. (Hint: K is allowed to take a decimal value. You will need the values from part c. If you couldn't solve part c, use expected value of the total ounces be 6,000 and the variance 700,000).

e) Approximate P(K ≥ 2 7). You may leave your answer in terms of the CDF of the standard normal, denoted Φ(x).

f) The next day, your assistant reports that 2.7 kegs were consumed on Friday, with 200 pints and 40 pitchers accounted for in receipts. Is it time to have a serious conversation with your staff?