110 Statistics B. Srinivasan Handout #35
Autumn 20062007 Out: Nov. 3, 2006 Due: Nov. 10, 2006
Homework #6
Reading: Rice 7.1-3,9.1-3 Submit in the inbox outside Sequioa 229 before 5:00 PM. In this sixth problem set, you will begin applying the theoretical machinery we have developed to real data sets. 1. This Is Not A Drill Here are some quick drill problems to get you back into the problem set mode. Each of these should only take you 15 minutes or so at a maximum. (a) (Adapted from Rice 1.8.73) A system has n independent units, each of which fails with probability p. The system fails only if k or more of the units fail. What is the probability that the system fails? Work this out numerically for a system with n = 1000 units, p = .002, and k = 3. (b) (Adapted from Rice 1.8.75) A population starts with one member. At time t = 1 it either divides with probability p or dies with probability 1 - p. If it divides, then both of its children behave independently with the same two alternatives at time t = 2. What is the probability that there are no members in the fourth generation? Numerically find the value of p that makes this probability equal to .1. This is a simple example of a branching process, a probabilistic model that is used for applications ranging from population modeling to calculations of the critical mass for a nuclear reaction. You may find it useful to use conditional probability. (c) (Adapted from Rice 1.8.76) The study of the capacities and waiting times of queues is extremely important for a number of applications ranging from customer service (Dell) to just-in-time manufacturing (Honda) to efficient internet video delivery (Akamai). Suppose we have a simple discrete time queue. At each unit of time the first person in the queue is served with probability p and independently a new person arrives with probability q. Note that if no people are in the queue, no service happens though arrivals are still possible. Let N (t) be the random variable for the number of individuals in the queue at time t and suppose there are N (0) = N0 people in the queue at t = 0. Define appropriate Bernoulli random variables Xt and Yt for departures and arrivals at time t to define N (t) in terms of N (t - 1). What is the average value of N (14) if N (12) = 1, p = .5 and q = .4? (d) SAT scores in the US used to be centered at 1000 with a standard deviation of about 200, a max of 1600, and a min of 400. Call this random variable X and assume that it can be approximated with a normal distribution with mean = 1000 and standard deviation = 200. Standardize this normal variable into a z-score and use the pnorm function in R to calculate the probability that X lies within 1 SD of the mean. Repeat this for 2 and 3 SDs respectively. Why would we want P (|X - | > 3) to be low for this application? 1
2. Taxes May Be Certain, But Death is a Random Variable (a) (Adapted from Rice 1.8.63) Suppose that the probability of living to be older than 70 is empirically found to be .6 and the probability of living to be older than 80 is .2. Express these probabilities in terms of a cdf for a life expectancy random variable T . (b) What is the conditional probability of surviving to 80 given that someone has survived to 70? (c) Now suppose we considered modeling T by an exponential random variable whose units are in years with parameter . Find the which gives a probability of .2 of living to be older than 80 years under this model. Use this to calculate the probability of living to be older than 70 years. Does this give a result which is consistent with the data? (d) Now use this same exponential model to calculate the conditional probability of surviving to 80 given that we have survived to 70. Repeat this for the conditional probability of surviving to 90 given that we have survived to 80. (e) The exponential model assumes a death rate which is constant with age. Describe qualitatively what modification of this model would be more realistic. 3. The Real Paradox: Why Aren't More People Watching The Family Guy? Simpson's paradox refers to the in phenomenon which the sign of the association between two random variables X and Y is reversed when conditioning on a third variable Z. This is related to the idea of conditional independence and dependence; just as conditioning on a third variable Z can make two independent variables conditionally independent P (X, Y |Z) = P (X|Z)P (Y |Z) (or vice versa), it can also change the sign of the dependence between X and Y . (a) Consider the following example from baseball. Player 1 is David Justice and Player 2 is Andy Van Slyke; the below table contains a subset of their statistics for the 1989 and 1990 seasons. 1989 1990 P1_Hits P1_At_Bats P2_Hits P2_At_Bats 12 51 113 476 124 439 140 493
Let X and Y be rvs for the batting averages of players 1 and 2, and let Z be the year. Calculate X - Y for each individual year and for the aggregate data set. What happened? (b) Here is the Berkeley admissions data: Dept_1 Dept_2 Dept_3 Dept_4 Dept_5 Dept_6 App_M Admit_Rate_Men App_F Admit_Rate_F 825 0.62 108 0.82 560 0.63 25 0.68 325 0.37 593 0.34 417 0.33 375 0.35 191 0.28 393 0.24 373 0.06 341 0.07
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Calculate the aggregate admit rates for men and women and compare them to the departmental admit rates. What is the Z variable in this problem? 4. Quitters Never Win and Winners Never Quit Suppose that Trent Johnson, the coach of Stanford basketball, is screening a player of unknown ability. He wishes to investigate this player's free throw shooting ability and initially models it with the prior Unif[0, 1]. The coach instructs the player to keep shooting until he makes a basket. Let N be the number of shots until his first basket. (a) Calculate the posterior distribution P (|N ). (b) Find a general expression for the value of which maximizes this conditional density function. (c) What is this value if N = 7? Does this seem like a good free throw shooter? (d) Now, AFTER the first round of free throw shooting, suppose the coach decides to give the player a second chance. He tells him to shoot exactly 10 more free throws and records the number M which are successful. Suppose further that the player now sinks M = 9 out of 10 of these free throws. What is the updated conditional distribution for in light of this new information? (HINT: Think of this as a new prior distribution...) (e) What is the new which maximizes the conditional density of ? (f) Use R to plot P (|N = 7, M = 9), P (|N = 7), and P () on the same axes. You will find week3_lecture7.R useful as an example of how to accomplish this in R . 5. Simulation of Summation We will use R to investigate the behavior of sums of random variables. (a) Let U1 , ..., U12 be uniform random variables between 0 and 1. Use a large number of samples with runif and hist (with the breaks=fd option) to look at the empirical density of U1 + U2 . (b) Now calculate the analytical density for S2 = U1 +U2 and overplot it on the original histogram. You will find week3_lecture7.R useful for overplotting examples. (c) Repeat part a) for S12 = U1 + ... + U12 . Make a histogram of the raw values of S12 with the same options as before. (d) Convert the values of S12 into z-scores by subtracting the mean and dividing by the standard deviation. Make a new histogram from these z-scores. (e) Overplot a standard unit normal distribution over the histogram of z-scores. Is the fit good? 6. Lake Wobegon and Lake Lagunita Suppose that we model IQ as X N (100, 152 ). (a) What is the average IQ of a place where everyone is above average? (That is, everyone is above the general population's average.).
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(b) What is the 90th percentile IQ of a place where everyone1 is above the 90th percentile? (c) Convert the preceding two values into the number of standard deviations from the population mean. For example, an IQ of 100 would correspond to 0 standard deviations above the population mean. (d) Now convert them into population percentiles. For example, an IQ of 100 would correspond to the 50th population percentile.
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See www.stanford.edu/dept/uga/applying/extras/1_2a6_profile.html#freshmen, for comparison.
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