1. Show that, for any non-negative random variable X, EX+E+≥2, X E max X. 21.
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- If a binomial experiment has probability p success, then the probability of failure is ____________________. The probability of getting exactly r successes in n trials of this experiment is C(_________, _________)p (1p)Assume that the probability that an airplane engine will fail during a torture test is 12and that the aircraft in question has 4 engines. Construct a sample space for the torture test. Use S for survive and F for fail.Suppose the probability of erroneously transmitting a single digit is P=0.03. Compute the probability of transmitting a 4-bit code word with (a) at most one error, and (b) exactly four errors.
- Of those women who are diagnosed to have early-stage breast cancer, one-third eventually die of the disease. Suppose a screening program for the early detection of breast cancer wras started in order to increase the survival rate p of those diagnosed to have the disease. A random sample of 200 women was selected from among those who were screened by the program and who vwere diagnosed to have the disease. Let x represent the number of those in the sample who survive the disease. A USE SALT (a) If you wish to determine whether the community screening program has been effective, state the alternative hypothesis that should be tested. O Hgip + O H;: p > O H: p= O Hgi p O Ho: P = O Hoi p * (c) If 194 women in the sample of 200 survive the disease, can you conclude that the community screening program was effective? Test using a = 0.05. Find the test statistic and the rejection region. (Round your answers to two decimal places. If the test is one-tailed, enter NONE for the unused region.)…2. Let X be a random variable with p.m.f., (1+ x? f(x) = -,x = -1,0,1,2,3 20 0 ,otherwise. Find E(3X2 + 6).Let X be a random variable from a Chi-Square Distribution with 13 degrees of freedom. Tatiana is interested in the random variable Y - aX + b. She knows that E(Y) = 24 and V(Y) = 416. What are all possible values of a and b that satisfy this criteria? Write your answer as an ordered pair, (a, b). If there are multiple solutions, separate the ordered pairs by а comma.
- Suppose that X and Y are independent random variables with EX=EY=100, and E[min(X,Y))=79. Find ElX-Y|, (the mean of the absolute value of X-Y).Let X be a random variable from a Chi-Square Distribution with 13 degrees of freedom. Jada is interested in the random variable Y = aX + b. She knows that E(Y) = 26 and V(Y) = 416. Jada wants to find out all possible values of a and b that satisfy this criteria. Write your answer as an ordered pair, (a, b). If there are multiple solutions, separate the ordered pairs by a comma. Hint: Use properties of expected value and variance to find a and b.Suppose that the random variables X and Y are independent with Var(X)=8 and Var(Y)=6. Calculate Var(5X−7Y+17)
- Many people believe that the daily change in price of a company's stock in the stock market is a random variable with mean 0 and variance o² during the middle 2 months between 2 earning report dates when there is not much news released. That is, if Yn represents the closing price of the stock on the nth trading day of that period, then Yn = Yn-1+Xn, for 1 ≤ n ≤ 60, where X₁, X2,..., X60 are independent and identically distributed random variables with mean 0 and variance o². Suppose that the stock price at the beginning of the 2-month period is 100. If o = 2, what is the approximate probability that the stock's closing price on the 30th trading day will exceed 110, i.e., what is P(Y30 > 110)Suppose that X₁, X₂, Xn and Y₁, Y2, . Yn are independent random samples from populations with means ₁ and ₂ and variances of and o2, respectively. Show that X - Y is a consistent estimator of μ₁ - 2.Suppose that X1, ..., Xn are random variables such that the variance of each variable is 1 and the correlation between each pair of different variables is 1/4. Determine Var (X1 + ...+Xn).
