6. The quality control manager of Mr. Gatti’s Pizza is inspecting slices of pepperoni pizza that just came out of the oven. If the production process is in control, the mean number of pepperoni pieces per slice is 4.0. What is the probability that in any particular slice being inspected: (a) fewer than five pepperoni pieces will be found? (Round answer to four decimal places.) (b) exactly five pepperoni pieces will be found? (Round answer to four decimal places.) (c) five or more pepperoni pieces will be found? (Round answer to four decimal places.) (d) either four or five pepperoni pieces will be found? (Round answer to four decimal places.)

A college placement office conducted a survey of 100 engineers who had graduated from Stanford University. For these engineers, the mean salary was computed to be $72,000 with a standard deviation of $22,000.  The distribution of salary is roughly bell shaped.

a) What percentage of these engineers will earn between $55,040 and $88,960?

b) What is the probability that the average income of 4 engineers will be between $55,040 and $88,960?

c) What would be the 90^thpercentile for the income of these individual engineers?

d) What would be the 90^thpercentile for the average income of groups of 9 engineers?

e) Why is the 90^thpercentile in (d) a smaller number than the value in (c)? Explain using the bell curve and what happens to it when you are looking at the average of a group.

1) Let U1, U2, ... be independent random variables, each uniformly distributed over the interval (0, 1]. These random variables represent successive bigs on an asset that you are trying to sell, and that you must sell by time = t, when the asset becomes worthless. As a strategy, you adopt a secret number \Theta and you will accept the first offer that's greater than \Theta . The offers arrive according to a Poisson process with rate \lambda = 1. For example, you accept the second offer if U1 <= \Theta and U2 > \Theta . What is the probability that you sell the asset by time = t? What value for \Theta maximizes your expected return?

2) Stochastic

A government study concluded that 45% of people receiving Social Security payments also receive retirement income from a private IRA. We decided to take a sample of 200 people receiving Social Security payments and asked them if they also receive retirement income from a private IRA. Define the random variable x to be the number of people who also receive retirement income from a private IRA.

Showing your work in Excel as demonstrated in class, what is the probability that x is:

Exactly 100?

Exactly 85?

Greater than or equal to 80?

Greater than 80?

Between 78 and 83, inclusive?

Between 100 and 200, inclusive?

1. Let X be a random variable with probability density function fX given by fX(x) = γαγ/ (x + α)^γ+1 , x ≥ 0,

0, x < 0,

where α > 0 and γ > 0.

(a) Find the cumulative distribution function (cdf) FX of X.

(b) Let Y = log(X+α /α) . Find the cdf of Y and identify the distribution.

(c) How could a realisation of X be generated from an R(0,1) random number generator?

(d) Let Z = min(X,M), where M > 0 is a fixed constant. Derive the cdf FZ of Z and compute its mean.

Question 4: (In this problem first find the probability by using SPSS and copy the output and then calculate the number of trees manually by using the probabilities.)

A certain variety of pine tree has a mean trunk diameter of μ= 150 cm, and a standard deviation of σ= 30 cm which is normally distributed. A certain section of a forest has 500 of these trees. Find Approximately

1. how many of these trees have a diameter smaller than 120

2. how many of these trees have a diameter greater than 160

3. how many of these trees have a diameter between 130 and 160.

4. how many of these trees have a diameter between 120 and 140.

Please answer the notes question: Exercise 4.4. Liz is standing on the real number line at position 0. She rolls a dic repeatedly. If the roll is 1 or 2, she takes one step to the right (in the positive direction). If the roll is 3, 4, 5 or 6, she takes two steps to the right. Let X,n position after n flips of the coin. Estimate the probability that X9o is at least 160. be Liz's.

Notes: Is this binomial???? if it is binomial distribution, then what is the P??? Bin~(n,p) If you don't answer this question and just finish the question above, you will get thumbdown

Second: is E(X^2)= 3* 90=270???

When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 10%. Let X = the number of defective boards in a random sample of size n = 25, so X ~ Bin(25, 0.1). (Round your probabilities to three decimal places.)

(a) Determine P(X ≤ 2).

(b) Determine P(X ≥ 5).

(c) Determine P(1 ≤ X ≤ 4).

(d) What is the probability that none of the 25 boards is defective?

(e) Calculate the expected value and standard deviation of X. (Round your standard deviation to two decimal places.)

When circuit boards used in the manufacture of compact disc players are tested, the long-run percentage of defectives is 10%. Let X = the number of defective boards in a random sample of size n = 20, so X ~ Bin(20, 0.1). (Round your probabilities to three decimal places.)

(a) Determine P(X ≤ 2).


(b) Determine P(X ≥ 5).


(c) Determine P(1 ≤ X ≤ 4).


(d) What is the probability that none of the 20 boards is defective?


(e) Calculate the expected value and standard deviation of X. (Round your standard deviation to two decimal places.)

The distribution of the number of viewers for the American Idol television show follows a normal distribution with a mean of 32 million with a standard deviation of 4 million. What is the probability next week's show will:

Have between 36 and 43 million viewers? (Round your z-score computation to 2 decimal places and final answer to 4 decimal places.)

Have at least 29 million viewers? (Round your z-score computation to 2 decimal places and final answer to 4 decimal places.)

Exceed 43 million viewers? (Round your z-score computation to 2 decimal places and final answer to 4 decimal places.)