4. The probability distribution of a random variable X is given below. X -2 0 1 3 5 P(X - x) 0.5 0.2 0.2 0.1 (a) Find the probability that X is larger than 2. (b) Find the expected value of X, E(X). (c) Find the variance of X, Var(X). 5. A test is taken by some students. Assume that the marks follow a normal distribution with a mean of 60 and a variance of 25. (a) Calculate the probability that the mark of a randomly selected student is greater than 58. A random sample of size n = 5 is selected. (b) Calculate the probability that the sample mean lies between 59 and 61. 6. Household expenditure in country A is normally distributed with a mean of $1,200 per week and a standard deviation of $400 per week. In country B it is also normally distributed but with a mean of $960 per week and a standard deviation of $200 per week. Which country has a higher proportion of households spending less than $800 per week? 7. Suppose that X is a normally distributed random variable with a mean of 0 and a variance of 1. (a) Find the probability that X + 4 is less than 4. (b) Find the value of b so that the probability of X -b being less than zero is 0.975