To Lecture Notes

IT 223 -- 2/16/11

Review Questions

1. What is the regression fallacy (also called regression to the mean)?

   Ans: Regression to the mean says that, given a pre-test/post-test situation, someone that does above average on the pre-test will do worse on the post-test; someone that does below average on the pre-test will do worse on the post-test.

2. What is the RMSE and how do you compute it?

   Ans: The root mean squared error (RMSE) is the standard deviation of the residuals. It can be computed as RMSE = SD_y √(1 - r²).

3. The heights and weights of a population of female models are recorded and descriptive statistics are computed:
   x = 1.7 m   SD_x = 0.05 m   y = 50 kg   SD_y = 5 kg   r = 0.8

   The heights and weights are bivariate normal.
   1. About what percentage of the models have weights over 58kg?

      Ans: Compute z = (y - y) / SD_y = (58 - 50) / 5 = 1.6. Look up -1.6 in the standard normal table: 0.0548 = 5.5%.

   2. Find the regression line for predicting weight from height.

      Ans: y = 80x - 86.

   3. Find the RMSE.

      Ans: RMSE = SD_y √(1 - r²)i 5 √(1 - 0.8²) = 5 * 0.6 = 3.

   4. Of those models that have a height close to 1.8m, what percentage of them have weights over 58kg.

      Ans: z = (y - y^{^}) / RMSE = (58 - 58) / 5 = 0. 50% of the models have a weight over z = 0.

4. Define these terms:
   1. Sample Space   Ans: The set of possible outcomes of a probability experiment.

   2. Probability Function   Ans: A function that assigns a number between 0 and 1 to each outcome in the sample space. The sum of the probabilities for the entire sample space must be 1.

   3. Random Variable   Ans: The process of choosing a random number. The more technical definition of a random variable is a probability function where all of the outcomes are real numbers.

5. What is wrong with this argument? Either the Cubs will win the World series or they won't, so the probability that they will win the world series is 50%.

   Ans: Every event has a probability assigned to it. Likely events have high probabilities; unlikely events have low probabilities. Just because you don't know the probability of an event does not mean that it is 50%.

6. Find as many errors as you can with this probability distribution?

   Outcome	Probability
   3	60%
   5	30%
   6	0%
   7	-10%
   9	110%

   Ans: Probabilities cannot be negative; probabilties cannot be more than 100%; the probabilities in the table must sum to 100%.

7. Is a "double down after a loss" a good idea in gambling? This means that every time you loose, you double your bet; when you win, you recover your losses. Then you start over with your normal bet after you win.

   Ans: This looks good, except for thing. If you lose too many times in a row, you will either run out of money or will not be able to double because you have reached the betting limit of the casino. This can easily wipe out all of your winnings in one play.

8. What is a Bernoulli random variable?

   Ans: A random variable whose only possible outcomes are 0 (with probability 1-p) and 1 (probability p). This can be expressed in a table like this:

   x	P(x)
   0	1-p
   1	p

9. What are three ways to obtain probabilities?

   Ans: Theoretical or a priori, empirical, subjective.

10. A bookie offers 15 to 1 odds for the Cubs to win the World Series in 2009. If this is a fair bet, what is the probability that the Cubs will win the World Series?

    Ans: 15p + (-1)(1 - p) = 0; 15p - 1 + p = 0; 16p = 1; p = 1/16 = 0.0625 = 6.25%.

Expected Value

• The expected value of a random variable is its theoretical average, based on the probabilities of its outcomes.

• The expected value of a random variable x can be calculated using the probability table.

  x     Probability x1 p1 x2 p2 ... ... xn pn

  Expected value = E(x) = x1 p1 + x2 p2 + ... + xn pn

• The expected value is the weighted average of the outcomes of the random variable. You don't need to divide by the sum of the weights because the sum of the probabilities is 1.

• The expected value is also called the risk.

  Example 10:   Suppose the probability that the Bears will the Super Bowl next year is 1 / 5 = 0.2. You make a bet with your friend. If the Bears will you get $100. If the bears do not will, you pay him $30. What is the expected value of this bet?

  Payoff	Probability
  -30	0.8
  100	0.2

  Note: the payoff of 30 in the table is negative because you are losing $30.

  Expected value = (-30) × 0.8 + 100 × 0.2 = -0.4

  The conclusion is that you will lose 40 cents every time you make this bet.

  Example 11:   Flip a coin 3 times. You win $10 every time 3 heads come up, you lose $1 every time 1 head comes up and you lose $5 every time 0 heads come up. Do do not lose or win anything if 2 heads come up. Compute the expected value.

  Ans: Here is the payoff table:

  Payoff	Probablity
  10	1/8
  0	3/8
  -1	3/8
  -5	1/8

  E(x) = 10(1/8) + 0(1/8) + (-1)(3/8) + (-5)(1/8) = 2/8 = $0.25

  Example 12:   A tropical island has the same probability distribution for the amount of rain every day:

  Rainfall	Probability
  0	0.3
  1	0.4
  2	0.2
  3	0.1

  What is the expected value for the amount of rain in one day?

  Ans: E(x) = x₁ p₁ + x₁ p₁ + x₁ p₁ + x₁ p₁

                = 0 × 0.3 + 1 × 0.4 + 2 × 0.2 + 3 × 0.1 = 1.1

  Example 13:   You pay $75 per year on a $500,000 house for insurance. The probability that your house is destroyed in a given year is 0.0001.

  1. What is the expected value for the insurance company per year?

     Ans: Here is the payout table:

     Payout	Probability
     75	0.9999
     -500,000 + 300	0.0001

     The expected value is 75 × 0.9999 + (-500,000 + 75) × 0.0001 = 75 × (0.9999 + 0.0001) + (-500,000) × 0.0001 = 75 + (-50) = $25.

  2. What is your risk per year?

     Ans: You pay $75 to the insurance company whether or not your house is destroyed, so the expected value is $-75.

  3. What would your risk be if you did not have insurance?

     Ans: The payout table is

     Payout	Probability
     0	0.9999
     -500,000	0.0001

     The expected value is 0 × 0.9999 + (-500,000) × 0.0001 = -$50.

  4. What would the "pure premium" to the insurance company be? The pure premium is the premium you would have to pay if the expected value of the insurance company were $0.

     Ans: Solve for p in the following: p (0.9999) + (-500,000 + p) 0.0001 = 0. This gives p = 50.

The Multiplication Rule for Independent Events

• Two events A and B are independent if the occurence of A does not affect the probability that B occurs.

• Example 14:   Flip a coin twice. Define the event A to be getting a head on the first toss. Define the event B to be getting a head on the second toss. The events A and B are independent.

• The Multiplication Rule:   If A and B are independent, then
  P(A and B) = P(A) × P(B)

• Example 15:   Flip a coin 10 times. What is the probability of getting all heads?

  Ans: P(H and H and ... and H (10 heads)) = P(H) × P(H) × ... × P(H) = 0.5¹⁰ = 0.0009766.

• Example 16:   The probability that I will die in a single plane ride is 0.0001. What is my probability of dying in 500 independent plane rides?

  Ans: Let A_i be the event of not dying in the ith plane flight. Then P(A₁ and ... and A₅₀₀) = P(A₁) × ... × P(A₅₀₀) = 0.9999⁵⁰⁰ = 0.951227. Therefore the probability of dying in one of the 500 flights is 1 - 0.9512 = 0.04877 = 4.9%.

  Working from the formula gives us
  1 - (1 - p)ⁿ = 1 - (1 - 0.0001)⁵⁰⁰ = 0.04877 = 4.9%

• Example 17:   The probability of contracting AIDS from a single sexual encounter is about 1 out of 1000. What is the probability of contracting AIDS from 20 independent encounters?

  Ans: 1 - (1 - p)ⁿ = 1 - (1 - 1 / 1000)²⁰ = 0.01981 = 1.98%

• In the real world, determining if two events A and B are independent is difficult. There are many lurking variables that can cause A and B to occur or not to occur.