Chapter 3- Questions
Prove 𝑉𝑎r(Y) = 𝐸[y^2] − 𝜇^2 or Var(Y) = E(y^2) - [E(y)]^2
3.3 Pt.2 Notes
Prove 𝐸(∑𝑔𝑖(Y)) = ∑𝐸[𝑔𝑖(Y)]
3.3 Pt.2 Notes
Prove 𝐸(𝑎Y+𝑏) = 𝑎𝐸(Y)+𝑏 where a, b are constant
3.3 Pt.2 Notes
Prove 𝑉𝑎r(𝑎Y+𝑏)=𝑎^2𝑉𝑎r(Y)
3.3 Pt.2 Notes
See homework 3.3 pt.1 for MC book questions
3.3 Pt.2 Notes
Example 1: Use the relationships on the previous page to re-compute the expected value and variance in examples 3 and 4 for X and Z. (Work is important here.)
Homework 3.3 Pt.2
Example 2: A vacation insurance policy covers delays in plane travel. The number of hours delay, N, on a plane journey has the following distribution: See Homework 3.3 Pt.2 for table For a delay of n hours, the following compensation is paid: Calculate the mean compensation per person for people who buy this insurance.
Homework 3.3 Pt.2
Homework 3.3 Pt.2 for Book MC
Homework 3.3 Pt.2
Example 10: A fair die is rolled repeatedly. Let Y be the number of rolls that have a result other than "one" before the first "one" is rolled. What is the probability function for Y? What is the cumulative probability function for Y?
homework 3.2
Example 1: Flip coin 3 times (this is probability experiment) so HHH or THT are members of the domain of the random variable. Let Y be the number of heads (3 or 1). Y is a random variable. What are the possible values of Y?
homework 3.2
Example 2: You go to Las Vegas and begin to put quarters in a slot machine. Let Y be the number of quarters you play before your first win of any amount. Y is a random variable. What are the possible values of Y?
homework 3.2
Example 3: Consider flipping a coin three times. Let Y be the number of heads that occur. What is the probability distribution of Y? What properties need to be checked to verify that it is a legitimate probability distribution? MC 1: What is the probability that you will have exactly one head in three flips? MC 2: What is the probability that you will have one or less heads in three flips?
homework 3.2
Example 4: You go to Las Vegas and begin to put quarters in a slot machine. Assume that the probability of winning on a single play is 0.05 and that plays are independent.Let Y be the number of quarters you play before your first win of any amount. MC 3: What is the probability distribution of Y? MC 4: What is the probability that you use 10 coins or less before you win?
homework 3.2
Example 5: A clinical researcher is studying a fatal disease. The random variable Y is the number of the year following diagnosis in which the patient dies. Her research yields the following probabilities: (See homework 3.2 for table) MC 5: What is the probability that a person's year of death will be 3 years? MC 6: What is the probability that a person's year of death will be 2 years or less?
homework 3.2
Example 6: What are the cumulative distributions for examples 3 to 5 above?
homework 3.2
Example 7: (slightly different from Example 4, read carefully) You play a slot machine repeatedly. The probability of winning on a single play is 0.05 and successive plays are independent. Y is the number of unsuccessful attempts before the first win. Find the probability distribution and cumulative distribution of Y. The probability distribution of Y is... The cumulative distribution of Y is... MC 7: How many times would you need to play the slot machine in order to be sure that your probability of winning at least once is greater than or equal to 0.99?
homework 3.2
Example 8: The number of fires per week that a station responds to is modeled by a discrete random variable with: 𝑃(𝑁=𝑛)=0.65^𝑛×0.35 for 𝑛=0, 1, 2 ... MC 8: Given that at least 1 fire occurs during a week, what is the probability that at least 5 fires occur during the same week.
homework 3.2
Example 9: Twelve playing cards are face down in a row on a table. Exactly one card is an ace. The cards will be turned up one at a time from left to right. Let Y be the number of non-ace cards turned over before the ace is turned over. What is the probability function forY?
homework 3.2
Example 1: A large HMO studied the number of children in a given birth. The distribution was as follows: (See homework 3.3 for table) MC 1: Find the expected value of the number of children per birth.
homework 3.3 pt.1
Example 2: Let Y be the random variable for the number of unsuccessful plays before the first win on the slot machine where the probability of winning on a single play is 0.05. MC 2: What is the expected value of Y? What is the interpretation of the expected value of Y?
homework 3.3 pt.1
Example 3: An automobile insurance company does a study to find the probability for the number of claims that a policyholder will file in a year. Their study produces the following probabilities for, Y, the number of claims per year. (See homework 3.3 for Table) MC 3: Find the expected value of the number of claims per policyholder per year. This policy pays $1000 for each claim. Let X be the claim amount per person per year. MC 4: What is the expected value of X? The company has a yearly fixed cost of $100 per policyholder (as well as paying $1000 per claim). Let Z be the cost per policy to the insurance company. MC 5: What is the expected value of Z?
homework 3.3 pt.1
Example 4: Go back to Example 3 and find the variance and standard deviation for each of the variables using the "by-hand" formulas from above. (Work is important here.) MC 6: What is 𝑉ar[𝑌]? MC 7: What is 𝜎𝑌? MC 8: What is 𝑉[𝑋]? MC 9: What is 𝜎𝑋? MC 10: What is 𝜎𝑍^2? MC 11: What is 𝜎𝑍? (See homework 3.3 pt.1 for confusion on formating)
homework 3.3 pt.1
Example 5: A random variable Y has the following distribution: 𝑃(𝑌=𝑦)=𝑉−0.02𝑦 for𝑦=0, 1, 2, 3, 4 where a is a constant. MC 12: Calculate the probability that y lies within 0.5 standard deviations of the mean.
homework 3.3 pt.1
Example 6: A tour operator has a bus that can accommodate 20 tourists. The operator knows that tourists may not show up, so he sells 21 tickets. The probability that an individual tourist will not show up is 0.02 and is independent of all other tourists. Each ticket costs $50 and is non-refundable if a tourist fails to show up. If a tourist shows up and a seat is not available (i.e. a tourist gets "bumped" from the tour), the tour operator refunds the ticket price ($50) and pays the tourist an extra $50 to compensate for their inconvenience. MC 13: What is the expected value of the revenue of the tour operator?
homework 3.3 pt.1
