Sup 16. Please help

1. Suppose five exams are to be scheduled. There are three possible slots per day when an exam can be held, and six possible days. There can be at most one exam per slot. a. In how many different ways can the five exams be scheduled over the slots in the six days ? [5 marks] b. In how many different ways can the five exams be scheduled so that each falls on a different day. [5 marks] C. What is the probability that the five exams fall onto different days ? Explain how you arrived at your answer using the axioms of probability. [5 marks] d. State the definition of conditional probability. [5 marks] Suppose the first exam is scheduled for the first slot of day one. Conditioned on this, what is the probability that the second exam is also scheduled on day one ? [5 marks] 2. On a given day of the week the weather can be either rainy or dry. Suppose the weather on each day of the week is independent of the weather on the other days of the week and that on a given day it rains with probability 0.1. a. State the definition of independence of random events. [5 marks] b. Using independence of the weather on each day calculate the probability that it rains only one day in a given week (i.e. out of seven days) ? [5 marks] c. What is the probability that it rains four days in a week? [5 marks] d. Let X be a random variable whose value equals the number of rainy days in a week. i. Derive the expected value E[X]. Hint: use the linearity of the expectation. [5 marks] ii. Derive the variance var(X). Hint: use the linearity of the variance of independent random variables. [5 marks] e. Write a short matlab program that draws random values for X. [5 marks] 3. Suppose a continuous random variable X has probability density function f(x)=0 when *50 and x21 and f(x)=1 when 0