Q1. According to the Statistical Abstract of the United States, there are 31.1 millions one-person households, 38.6 millions two-person households, 18.8 millions three-person households, and 27.5 millions four or more person households.
a) Construct a table of each value of the random variable X, the number of persons in the household, and its estimated probability or frequency.
b) Compute and interpret P(X = 3) and P(X > 4).
c) Find the mean, variance and standard deviation for the population of the number of persons per household.
Q2. Consider the following Table:
Y 0 1 2 3
P(Y) 0.10 0.30 0.40
a) Complete the Table with the probability of P(X = 3).
b) Calculate the mean, variance and standard deviation of Y.
c) Suppose now we have a new random variable X = 2 + 3Y. Construct a Table with the probability distribution of X for each value of Y and X.
d) Calculate the mean, variance and standard deviation of X.
e) Use the properties of the expected value and variance to obtain the expected value of X, E(X), and the variance of X, V(X). Compare your answers.
Q3. David is an insurance broker who believes that the probability of making a sale is 0.3.
a) Find the mean and variance of a random variable defined to be 1 it he makes a sale and 0 otherwise.
b) What is p and 1 - p? Check that the probability distribution satisfies the two conditions discussed in class.
c) Suppose now David has 5 contracts and now the probability of making a sale is 0.4. Compute the probability that he makes:
i. at most one sale,
ii. between 2 and 4 sales.
d) Graph the distribution function checking that the probabilities are between 0 and 1 for all x, and that the sum of the probabilities is equal to 1.
Q4. University H has decided to accommodate more students this Fall. Knowing that typically 40% if the students admitted actually enroll, answer the following questions using Table in Keller 2012 or Excel (please see Keller 2012, page 248):
a) What is the probability that at most 6 students will enroll if the University H. offers admission to 10 new students?
b) If admission is offered to 20 students, what is the probability that more than 12 students will enroll?
c) Assume that the probability increases to 70%. What is the probability that at least 12 out of 15 students will actually enroll?
Q5. Assume that Charlotte has two stocks: Disney (D) and Amazon (A). For each of these stocks, the possible percent returns are {0%, 5%, 10%, 15%}. The joint probability of P(d, a) = 0.0625 for all values of D and A.
a) Construct a Table with joint probabilities.
b) Find the marginal probabilities.
c) Are Disney and Amazon stock returns independent?
d) What is the mean and variance of the stocks D and A?
e) What is the covariance between A and D?