4.34. Find the moment-generating function of the dis- crete random variable X that has the probability distri- bution (51) 20 f(x) = 2 and use it to determine the values of u and 2. for x = 1,2,3,... 3
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I know that Mx(t) = summation of (etx times f(x)), but I wasn't sure what to do after that.
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- If X1 and X2 constitute a random sample of size n = 2from an exponential population, find the efficiency of 2Y1relative to X, where Y1 is the first order statistic and 2Y1and X are both unbiased estimators of the parameterConsider two securities, the first having μ1 = 1 and σ1 = 0.1, and the secondhaving μ2 = 0.8 and σ2 = 0.12. Suppose that they are negatively correlated,with ρ = −0.8. Denote the expected return and its standard deviation as functions of π byμ(π ) and σ (π ). The pair (μ(π ), σ (π )) trace out a curve in the plane as πvaries from 0 to 1. Plot this curve in R.The critical value for t for n = 10 for a one tail test at α = 5% LOS is
- Let the following simple random sample following:1. Binomial pmf. (11, ¾);2. Uniform pmf;3. Uniform pdf (0, a);4. Exponential pdf with (µ) .Find the corresponding pmf/pdf of Y1 , Y4 , Y7 and F(Yi) whereY1<Y2<...<Y119.1) Suppose X1, X2, and X3, denotes a random sample from the exponential distribution with density function shown in the image. a) Which of the above estimators are unbiased for θ? b) Among the unbiased estimators of θ, which has the smallest variance?Let m(t) be the moment generating function of a random variable X. Show that the random variable W = 10X is m(10t). What is the moment generating function of Z = X-5 in terms of m(t)?
- .A sample of 9 measurements, randomly selected from a normally distributed population, resulted in x= 2.6, and s= 0.9 Conduct a hypothesis test to verify the claim that the population mean is greater than 2.5 . Use a=.05In recent years, several companies have been formed to compete with AT&T in long-distance calls. All advertisethat their rates are lower than AT&T's. AT&T has responded by arguing that there will be no dierence in billingfor the average consumer.Suppose that the average bill for current AT&T customers is $21. A statistician takes a random sample of 100customers and recalculates their last month's AT&T bill using the rates quoted by a competitor. Let X be therandom variable for the amount of a recalculated bill, and µ be the unknown population mean of X. The samplemean and standard deviation for X are $20 and $6. The statistician wants to test if µ is signi cantly dierent from$21, which is the average bill for current AT&T customers (d) Using 5% signi cance level, calculate the critical values of the test. Is the null hypothesis rejected?(e) The competitor would react by using dierent set of hypotheses. For example, consider H0 : µ = 19 andH1 : µ 6= 19. What…In recent years, several companies have been formed to compete with AT&T in long-distance calls. All advertisethat their rates are lower than AT&T's. AT&T has responded by arguing that there will be no dierence in billingfor the average consumer.Suppose that the average bill for current AT&T customers is $21. A statistician takes a random sample of 100customers and recalculates their last month's AT&T bill using the rates quoted by a competitor. Let X be therandom variable for the amount of a recalculated bill, and µ be the unknown population mean of X. The samplemean and standard deviation for X are $20 and $6. The statistician wants to test if µ is signi cantly dierent from$21, which is the average bill for current AT&T customers.(a) The statistician wants to formulate hypotheses in favor of AT&T. State the hypotheses.(b) Calculate the t-statistic. Under what signi cance level, among 1%, 5% and 10%, is the null hypothesis rejected?(c) Calculate the…
- A researcher is using a two-tailed hypothesis test with α = 0.01 to evaluate the effect of a treatment. If theboundaries for the critical region are t = ± 2.845, then how many individuals are in the sample?A. n = 23B. n = 22C. n = 21D. n = 20E. cannot be determined from the information givenIn the tt-test of significance for the hypotheses H0:μ=11 and Ha:μ<11, the value of the test statistic was found to be t = 1.205 using a sample of size 18. Estimate the upper bound for the P-value associated with this test. That is, fill in the blank P <P< _____. Please show work!Show that, if X1 and X2 are independent random variables with geometric distributions of aparameter p, then Y = X1 + X2 has a negative binomial distribution with parameters p andk = 2.
