Let be the Wwilson-adjusted sample proportion out of n - 10 trials. Suppose Y is the...
Problem 1 Consider a sequence of n+m independent Bernoulli trials with probability of success p in each trial. Let N be the number of successes in the first n trials and let M be the number of successes in the remaining m trials. (a) Find the joint PMF of N and M, and the marginal PMFs of N and AM (b) Find the PMF for the total number of successes in the n +m trials.
Problem 1 Consider a sequence...
Suppose that total 5 independent trials having a common probability of success 1/3 are performed. If X is the number of successes in the first2 trials, and Y is the number of successes in the final 3 trials, then X and Y are independent, since knowing the number of successes in the first 2 trials does not affect the distribution of the number of successes in the final 3 trials (by the assumption of independent trials). Find the joint p.d.f....
trial. Consider n trials , each with probabılity of success p. Assume the trials are independent given p. Now, suppose p ~Beta(α, β), 2-1, , n. Recall that if X is a Beta r.v r@ + β) Ta r"-1 (1-2)β-1I(0 < x < 1), x(x - (1 α > 0,3 > 0 αβ E(X) = (a) Compute the expected value of the total number of successes. (b) Compute the variance of the total number of successes.
7.75. Let us repeat Bernoulli trials with parameter 0 until k successes occur. If Y is the number of trials needed: (a) Show that the p.d.f. of Y is g(y; 0) Oyyk. k- k+1,..., zero elsewhere, where 0< es 1. (b) Prove that this family of probability density functions is complete. (c) Demonstrate that E[(k - 1)/(Y- 1)] 0 (d) Is it possible to find another statistic, which is a function of Y alone, that is unbiased? Why?
7.75. Let...
In the binomial probability distribution, let the number of trials be n = 3, and let the probability of success be p = 0.3634. Use a calculator to compute the following. (a) The probability of two successes. (Round your answer to three decimal places.) (b) The probability of three successes. (Round your answer to three decimal places.) (c) The probability of two or three successes. (Round your answer to three decimal places.)
You perform a sequence of m+n independent Bernoulli trials with success probability p between (0, 1). Let X denote the number of successes in the first m trials and Y be the number of successes in the last n trials. Find f(x|z) = P(X = x|X + Y = z). Show that this function of x, which will not depend on p, is a pmf in x with integer values in [max(0, z - n), min(z,m)]. Hint: the intersection of...
2. Let X~Bin(n, p) with n known. State whether the following expressions are statistics or not. If an expression is not a statistic, explain why. (a) The number of successes X observed in n trials The sample proportion of successes D (c) z -, where X ~ N(5,4) p-P (d)-p I-D
Suppose that in a series of n = 250 independent trials, with an unknown probability of success p, x = 95 “successes” were recorded. a) Test the null hypothesis H0 : p = 0.30, against the two-sided alternative H1 : p ≠ 0.30, at the confidence level α = 0.01. b) Give a 95% two-sided confidence interval for the unknown probability p. c) Suppose that the number of trials n can be determined before the random experiment was carried out....
1. The proportion of defective items in a large lot is p. Suppose a random sample of n items is selected from the lot. Let X denote the mumber of defective itens in the sample and let denote the number of non-defective items. (a) Specify the distributions of X and Y, respectively. Are they independent? (b) Find E(X-Y) and var(X Y).
1. The proportion of defective items in a large lot is p. Suppose a random sample of n items...
Suppose you want to estimate a particular population proportion p of “success”, say the proportion of Cal Poly students who plan to go to Coachella this year. Consider two methods of collecting data. 1) Select a simple random sample of size n for a fixed, specified n. Let X be the count of successes in the sample. For example, select a sample of n = 30 students, and say for the selected sample X = 3 students plan to attend...