Consider a set of n trials involving independent and identically distributed random variables X1;X2; : : : ;Xn; where the outcome of 1 corresponds to success and the outcome of 0 corresponds to failure, and the probability of success on each trial is p. (a) Show that E(Xi) = p, and Var(Xi) = p(1 p) for i = 1; 2; : : : ; n. (b) Let Y = X1 +X2 +: : :+Xn: Show that E(Y ) = np, and Var(Y ) = np(1p). (c) Let ^p = Y=n. Show that E(^p) = p, and Var(^p) = p(1 p)=n. Use facts from lecture notes and/or the Lohr Appendix handout to justify your reasoning for these problems.

Adifferent website gives the cost for the first train as £56.88 and the second train as £20.11 with a £9.50 charge for cycle storage for the whole trip. how much would the journey cost you?

Barbara has a good credit history and is able to purchase a car with a low-interest car loan. she co-signs a car loan for her friend jen, who has poor credit history. then, jen defaults on the loan. who will be held legally responsible by the finance company and why? select the best answer from the choices provided. a. barbara will be held legally responsible because she has a good credit history. b. jen will be held legally responsible because she drives the car. c. barbara will be held legally responsible because she is the co-signer. d. jen will be held legally responsible because she has a poor credit history. the answer is a

Damon makes 20 cups of lemonade by mixing lemon juice and water. the ratio of lemon juice to water is shown in the tape diagram. lemon juice\text{lemon juice} lemon juice water\text{water} water how much lemon juice did damon use in the lemonade? cups