1. Give brief answers to each of the following questions, providing a justification for eachpart. (a) Simplify: \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right) (b) For two independent events A and B such that P(A) > 0 and P(B) > 0, show that: P(A \cup B)<P(A)+P(B) (c) A fair die is thrown. Determine whether the events 'A = the die shows a total of 1, 2or 3' and 'B = the die shows an even score' are independent. \text { (d) If } X \text { is a random variable with } \operatorname{Var}(\sqrt{X})>0 \text {, explain why: } \mathrm{E}(X)>(\mathrm{E}(\sqrt{X}))^{2} (e) When sampling from a Poisson distribution with rate parameter A, state thesampling distribution of X according to the central limit theorem, for large n. (f) In your own words, briefly explain what a p-value is. (g) A 4-by-2 contingency table results in a test statistic value of 6.724. Is this significantat the 5% significance level? (h) X and Y are identically distributed random variables. Suppose Var(X) = 4 and the correlation between X and Y is p = -0.5. Derive the covariance of X and Y.

Fig: 1

Fig: 2

Fig: 3

Fig: 4

Fig: 5

Fig: 6

Fig: 7

Fig: 8

Fig: 9

Fig: 10

Fig: 11

Fig: 12