A corporation produces packages of paper clips. The number of clips per package varies, as indicated below for a sample of 20 packages.

Create a probability density function.

What is the probability that a randomly chosen package contains between 48 and 53 clips (inclusive) per package?

Please specify your answer in decimal terms and round your answer to the nearest hundredth (e.g., enter 12 percent as 0.12).

N people put their belongings in the jar one by one, and when they are picked one arbitrarily, the number of people who choose their own things is called a random variable X.

a. If the i-th person chooses his belongings, Xi = 1, otherwise Xi = 0, explain whether the probability variable Xi can have a Bernoulli distribution, and mark the probability variable X as a function of Xi.

b. Show E(Xi) = 1 / N and Var(Xi) = (N - 1) / N^2.

c. Find the expected value of E (X) and the distributed Var (X) for choosing your belongings.

For a certain river, suppose the drought length Y is the number of consecutive time intervals in which the water supply remains below a critical value y₀ (a deficit), preceded by and followed by periods in which the supply exceeds this critical value (a surplus). An article proposes a geometric distribution with p = 0.385 for this random variable. (Round your answers to three decimal places.)
(a) What is the probability that a drought lasts exactly 3 intervals? At most 3 intervals?

(b) What is the probability that the length of a drought exceeds its mean value by at least one standard deviation?

Three hats each contain ten coins. Hat 1 contains two gold coins, five silver coins and three copper coins. Hat 2 contains four gold coins and six silver coins. Hat 3 contains three gold coins and seven copper coins. We randomly select one coin from each hat.

(a) The outcome of interest is the color of each of the three selected coins. List the complete sample space of outcomes and calculate the probability of each.

(b) Let X be the number of gold coins selected. Find the probability distribution of X

An ordinary six-sided die is tossed once and one slip of paper is randomly drawn from a jar containing 5 slips, each of which is lettered U, W, X, Y, or Z. (a) Write the complete sample space that describes all possible outcomes. (b) Determine the probability that the die will show a 6 and that the drawn slip of paper is a U or W. (c) Let A represent the event that the die shows a number greater than 4. Let B represent the event that the drawn slip of paper is not Z. Determine the probability that at least one of these two events occur.

Each student arrives in office hours one by one, independently of each other, at a steady rate. On average, three students come to a two-hour office hour time block. Let S be the number of students to arrive in a two-hour office hour time block.

What is the distribution of S? What is its parameter?

Group of answer choices

S ∼ Geo(1/3)

S∼Pois(3)

S∼Bin(2,1/3)

S-Pois(1.5)

What is the probability that S = 4?

What is the probability that S ≤ 2?

What is the variance of S?

What is the expected value of S?

Suppose bit errors in a digital data file occur independently with probability p = 0.25 · 10^−6 per bit.

X = number of bit errors in a 1 Mbyte data file (= 10^6 bytes) Calculate exactly or with suitable approximation

(a) X's standard and standard deviation;

b) the probability that at least three bit errors occur in the data file.

c) Suppose that instead of an unknown parameter. A test file of size 3.85 Mbytes is checked where 10 bit errors are found. Enter a numerical point estimate of p and enter its Standard deviation d [ˆp].

For each of the following situations, explain whether the binomial distribution applies for X.

a. You are bidding on four items available on eBay. You think that you will win the first bid with probability 25% and the second through fourth bids with probability 30%. Let X denote the number of winning bids out of the four items you bid on.

b. You are bidding on four items available on eBay. Each bid is for $70, and you think there is a 25% chance of winning a bid, with bids being independent events. Let X be the total amount of money you pay for your winning bids.

2. Alice and 10 other users are sending packets using the pure ALOHA protocol. The duration of a packet is 40 msec. Each user sends a packet (including both originals and retransmissions) following a Poisson process with rate 1/400 packets/msec.

1. (a) What is the chance of success on Alice’ first attempt? (hint: the probability that there is no othertransmission within the vulnerable period of Alice’s attempt)

2. (b) What is the probability that Alice gets exactly k collisions and then a success? (hint: consider a Bernoulli trials process)

3. (c) What is the expected number of transmission attempts needed to successfully send a packet?