Question

andom variables Xi, .. . , Xn are iid with mean fu- 12 and variance ?2 = 5. Let X (X + Xn)/n denote the average and T- Xi+.. + Xn denote the total of the variables. For parts (a) through (d) below, calculate the required probability and state whether your calculation is exact or approximate (a) P(X> 13) for n 5 if the underlying distribution is normal (b) P(T < 300) for n -20 if the underlying distribution is gamma. (c) P (410<T< 475) for n = 35 if the underlying distribution is normal 11.7) if the underlying distribution is unknown for n E P (X 150, 100, 150) (d)

Answer 1

Random variables x -1 ...X _n are wth mean M = 12 and variance partial differentiation 5 let X^bar = (x + ...+x _n)/n denote the average and T = x _1 + + x_n denote the total of the variables a) the gien variables are iid and follows a normal distribution since the variables are normally distributed the sample mean will also follow a normal distribution with mean M and variance p(x^bar > 13) = 1 - p(x^bar < 13) p(x - m/partial differentiation/rootn > 13 - 12/root5/5) = 1 - p(z < 1) = 0 b) The given random variables are iid amd follows a gamma distribution