Question 1 [10 marks]Alice and Kim play a card game which can be either won or lost. The probability of Kimwinning 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 3red 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 washeads?Question 3 [8 marks]Customers arrive at a shop following a Poisson distribution with an average of one customerevery 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, includingthe values of the parameters.Question 4 [12 marks]Measurement of a blood test is a random variable X with cumulative distribution functiongiven bya. Find fX(x), the probability density functicard game作业代做、代写Python，c/c++语言作业、Java编程设计作业调试 帮做Java程序|代做SPSon.b. Graph fX(x).c. Find the mean and the variance of X.d. Find the median of X.2Question 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 randomsample 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 yourreason(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 befX,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 ).3Some special discrete distributionsDistribution Probability Function E(X), Var(X) and MX(t)Bernoulli p(x) = ( πx(1 π)�elsewhere�4Some special continuous distributionsDistribution Probability Density Function E(X), Var(X) and MX(t)Γ(n) = (n 1)!, n is positive integer.5Normal Cumulative Distribution Function, Φ(x) = P(Z ≤ x)转自：http://www.3daixie.com/contents/11/3444.html