card game代做、代写Python，c/c++语言、Java编程设计调试

日期：2019-05-29 11:08

Question 1 [10 marks]

Alice and Kim play a card game which can be either won or lost. The probability of Kim

winning is 0.2.

a. If they play 10 games:

i. What is the probability that Kim wins no games?

ii. What is the probability that Kim wins less than 3 games?

b. Suppose they play games sequentially, with no limit to the number of games:

i. What is the expected number of games until Kim wins a game?

ii. What is the expected number of games until Kim wins 3 games?

Question 2 [10 marks]

There are two bowls. Bowl A contains 1 green and 3 red balls. Bowl B contains 2 green and 3

red balls.

A fair coin is tossed. If a head then two balls are selected without replacement from Bowl A.

Otherwise two balls without replacement are selected from Bowl B.

a. What is the probability that one ball is red and one ball is green?

b. If one ball is red and one ball is green, what is the probability that the coin toss was

heads?

Question 3 [8 marks]

Customers arrive at a shop following a Poisson distribution with an average of one customer

every 20 minutes.

a. Compute the probability that no customers arrive during a 20 minute period.

b. Compute the probability that more than 3 customers arrive during an hour.

c. During 1 hour what is the expected number of customers to arrive?

d. Give the probability density function of the waiting time between customers, including

the values of the parameters.

Question 4 [12 marks]

Measurement of a blood test is a random variable X with cumulative distribution function

given by

a. Find fX(x), the probability density function.

b. Graph fX(x).

c. Find the mean and the variance of X.

d. Find the median of X.

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Question 5 [10 marks]

A computer manufacturer claims that only 2% of their computers are defective.

a. What is the exact probability of finding at most one defective computer in a random

sample of 100 computers?

b. Employ a Poisson approximation to solve part (a).

c. Employ a Normal approximation to solve part (a).

d. Which approximation (Poisson or Normal) is more appropriate in this case? Give your

reason(s).

Question 6 [10 marks]

Let X be a random variable with the following probability density function:

Using following relationship R ∞0

a. Show that fY (y) is a valid probability density function.

b. Show that the moment generating function MY (t) = 4(2t)

2 for t 6= 2.

c. Obtain the first and second raw moments.

d. Using these raw moments determine the mean and variance.

Question 7 [10 marks]

Let the joint probability density function of X and Y be

fX,Y (x, y) =c (x + y), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1,

0, otherwise.

a. Show that c = 1.

b. Find the marginal probability density functions fX(x) and fY (y).

c. Are X and Y independent? Give your reasoning.

d. Find E(X), E(Y ) and E(XY ).

e. Find cov(X, Y ).

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Some special discrete distributions

Distribution Probability Function E(X), Var(X) and MX(t)

Bernoulli p(x) = ( πx(1 π)

elsewhere

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Some special continuous distributions

Distribution Probability Density Function E(X), Var(X) and MX(t)

Γ(n) = (n 1)!, n is positive integer.

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Normal Cumulative Distribution Function, Φ(x) = P(Z ≤ x)