27. The SD of a bounded random variable. a) Let X be a random variable with 0

27. The SD of a bounded random variable. a) Let X be a random variable with 0 X 1 and E(X) = g. Show that: Gi) O Var(X) S /1(1 - p) [Hint: Use X2 b) Let X be a random variable with a X b and E(X) = Show that: Gi) o Var(X) (II - - g) (b- a)2; (iii) O S SD(X) (b- a)/2. , 9 is exactly 4! c) The standard deviation of a list of a million digits 0, 1, 2, How many nines are there in the list? Or is it impossible to answer this question without more information? 28. Let S be the number of successes in n independent Bernoulli trials, with possibly different probabilities m, ... , pn on different trials. Show that for fixed = E(S), Var(S) is largest in case the probabilities are all equal.