Question: (a) A random variable X has probability density function f(x) = 12x(1 x) 2 for 0 < x < 1 and f(x) = 0 otherwise.

(a) A random variable X has probability density function f(x) = 12x(1 x) 2 for 0 < x < 1 and f(x) = 0 otherwise. (i) Show that = E(X) = 2 5 and 2 = Var(X) = 1 25 . (ii) Show that Pr(X 1/2) = 11/16. (iii) Assume X1, . . . , X10 are independent and identically distributed with this probability density function. Corresponding to each Xi , define Yi = 1 if Xi 1/2 and Yi = 0 for otherwise. Find the probability Pr(P10 i=1 Yi > 2). (b) Suppose that a carrier of corona virus (without symptom) has the probability of p = 0.15 to pass the virus to a close contact. If this person has contacted 100 people in two days, use the Central Limit Theorem to approximate the probability that the number of people affected by this person is greater than 10 in these days.

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