YMS Chapter 7 Random Variables
Q1. A random variable is a variable whose value is a ______of a random phenomenon.
Q2. A random variable with a countable number of possible values is a _____ random variable.
Q3. What is a probability distribution of a discrete random variable?
Q4. For the probability distribution of a discrete random variable, every probability is between ___ and ___, and the sum of all the probabilities is equal to ___.
Q5. In a probability histogram, what quantity do the horizontal and vertical axes represent, respectively?
Q6. A continuous random variable can take on how many values for a certain interval in its domain?
Q7. A continuous random variable’s probability distribution is described by a graph called the ___.
Q8. Events, for continuous random variables, are described by the random variable’s taking on a value within a certain interval. The probability of that event is represented by what aspect of the density curve?
Q9. Suppose you have a continuous random variable X. What is the probability that X=10?
Q10. In a continuous probability distribution, what is the relationship between the probability that X<10 and the probability that X<=10?
Q11. True or false: the normal distribution is an example of a continuous probability distribution.
Q12. The mean of a discrete random variable is the sum of the products of all the possible values and the ______.
Q13. Suppose there are two possible outcomes for a certain random variable, 0 and 100. The probability of getting 0 is .99 and the probability of getting 100 is .01. What is the mean of the random variable?
Q14. The mean of a random variable is often called the e_____ v_____ of the variable.
Q15. Someone is offered a gambling game where there is a .25 chance of her losing $100, and a .75 chance of her winning $60. If she plays many times, what would her average winnings be?
Q16. Someone is invited to send in a contest entry in which the chances are 1 in 50 million of winning a million dollars, and one in a million of willing a thousand dollars. How do the expected earnings (or the mean earnings) from this contest compare with the price of a first-class stamp?
Q17. The mean of symmetric continuous probability distributions lies at the ____ of the curves.
Q18. The variance of a discrete random variable is the sum of the products of the squared deviation of each possible value from the mean of the distribution and the _____ for that value.
Q19. Suppose there is a distribution with possible values 0, 1, and 2, each with probability 1/3. What is the variance, i.e. the sigma-squared, of this distribution? (This is also known as the variance of the population.)
Q20. Think back to the definition of the variance of a sample. Suppose you had a sample consisting of 0, 1, and 2, with mean 1. Is the variance of this sample the same as the variance of the population?
Q21. Please take a few seconds to enter 0,1,and 2 in a list on your calculator. (On the TI-83 or 84, stat >edit.) Then please compute 1 variable stats on these. (Stat>calc>1varstats, listname). Look at sx and sigma x results. What are they, and why do they make sense vis a vis the different definitions of population and sample variance and standard deviation?
Q22. What is the law of large numbers, in your own words?
Q23. True or false: If by chance, you flip a coin and get 10 heads in a row, the law of large numbers tells us that if we flip many more times, we will get just a tiny bit under 50% heads in the remaining tosses, to compensate for the first 10 heads and make the long-range probability equal 50%.
Q24. True or false. When you are picking a “large number” so as to make the sample mean get within a certain distance of the population mean, you need a larger number the greater the variability of the random outcomes.
Q25. True or false: the mean of a linear function of a random variable is that same linear function of the mean of the random variable. In other words, the mean of a + bX is a +b*the mean of X.
Q26. The mean of the sum of two random variables equals what?
Q27. If the mean amount that Linda makes at her lemonade stand per day is $10 and the mean amount her brother Tom makes is $9, what’s the average of their total daily receipts?
Q28. Suppose someone tells you that the standard deviation of single scores on the SAT is 100 points. Suppose that there are two people who take the SAT independently of one another. How would you find the sd of the sum of their scores?
Q29. If the standard deviation of the SAT math and critical reading are both about 100, is the standard deviation of the sum of these two scores for an individual more or less than 141.42? Why?
Q30. Can you give an intuitive explanation for why the variance of the sum of two random variables is increased, the more highly they are correlated with each other?
Q31. A linear combination of two independent normally distributed random variables is distributed how?
YMS Chapter 8 The Binomial and Geometric Distributions
Q1. Suppose someone looks at the numbers of 1’s, 2’s, 3’s, 4’s, 5’s, and 6’s that result from 600 die rolls. Is this situation an example of the “binomial setting”?
Q2. What are the four requirements for the binomial setting?
Q3. The distribution of the number of successes out of n trials (with probability of success p on each trial) is the ______.
Q4. If someone has 51 socks in a drawer, with 1/3 red and 2/3 black, and the person grabs a handful of 5 of them, and counts the number of black, will the results of such a trial follow the binomial distribution? Why or why not?
Q5. Suppose you roll a die 1000 times and count “1” as success and “anything else” as failure. Is this an example of the binomial setting, and does the count have the binomial distribution? Why or why not?
Q6. In Chapter 1, the word distribution was defined as what values the variable takes and how often it takes these values. Let’s say you roll a die 1000 times and count the number of 1’s. The count of successes comes out to 165. Someone asks you, “What does this have to do with a distribution? We just got one number from this experiment. How’s anybody going to plot a histogram or any other representation of a distribution with this?” What would be your answer?
Q7. When we say a certain random variable has a B(100, .7) distribution, what do we mean?
Q8. If there is a discrete random variable (such as a binomial), and you want to find the probability of any given value of X, what function do you use – the cumulative distribution function or the probability distribution function? (cdf or pdf?)
Q9. Suppose you want to know the probability that a binomial random variable B(100, .7) takes on a value less than or equal to 60. One way would be to use the binomial pdf for values 0, 1, .. 60 and then add them all up. A much less laborious way would be to do what?
Q10. Suppose you roll a die six times, and you want the probability of getting exactly 3 1’s. What would be the appropriate expression of the binomial formula that would give the answer to this?
Q11. Can you please explain why the binomial probability formula is as it is, using this example of rolling a die six times? Please give an explanation for why each of the three factors is what it is.
Q12. Can you please explain why, to obtain the binomial coefficient, you use the number of combinations rather than the number of permutations, in calculating n choose k?
Q13. In chapter 7 we learned that means and variances are additive when you want to know the mean and variance of sums of independent random variables. How are these facts crucial in figuring out the formulas for the mean and variance of a binomial random variable?
Q14. What are the formulas for the mean, variance, and standard deviation of a binomial random variable in terms of n and p, and, if you want, q (or 1-p).
Q15. When n is “large,” the binomial distribution with n trials and success probability p can be approximated by what?
Q16. As a rule of thumb, the normal distribution may be used as an approximation to the binomial when both np and nq (expected successes, expected failures) equal or exceed what number?
Q17. Please describe how to have your calculator simulate a binomial experiment. What are the keys that you press?
Q18. For a binomial setting, the number of trials is fixed, and the random variable is the number of successes in that trial. For a geometric setting, the random variable is the number of ____ necessary to achieve the first _____.
Q19. True or false: in the geometric setting, as in the binomial setting, you have 1. two categories, 2. with the same probability for each observation, and 3. independent observations.
Q20. In a geometric setting, with probability of success p, what is the probability that the first success will occur on the nth trial?
Q21. True or false: the probabilities of success on the first, second, third, etc. trial in a geometric setting, when arranged in order, form a geometric series where p is the first term and each successive term being (1-p) (or q) times the previous one?
Q22. True or false: if you apply the formula a/(1-r) for the sum of the terms of an infinite geometric series, where a is the first term and r is the ratio of each term to the previous one, for the geometric setting p is the first term and (1-p) is the ratio, so the sum becomes p/(1-(1-p)) or 1. Thus even though there are infinitely many possibilities for the outcome of the experiment in the geometric setting, the probabilities of each outcome sum to 1.
Q23. If your chances of rolling a 1 on a die roll are one in 6, what is the expected or average number of times that you would have to roll the die before getting a 1?
Q24. If your chances of getting a success at anything in the geometric setting is p, what is the average or expected number of trials that you would have to conduct before getting a success?
Q25. What is the variance in the geometric random variable?
Q26. In the geometric setting, if q=1-p, what is the probability that it takes more than n trials to see the first success?
Q27. On page 470 of YMS there is a derivation of the formula for the probability that it takes more than n trials to see the first success. Can you think of a really simple way to arrive at the same formula?
Q28. For a geometric distribution, would you say that it is approximately true that 34% of the observations would fall between the mean and 1 standard deviation above the mean, and 34% would fall between the mean and 1 standard deviation below the mean? Why or why not?
Q29. Suppose that some experts estimate that the probability of a major nuclear war in any given year is 1%. You think that you will live another 65 years. You are wondering what the chance is that you will and will not see a nuclear war. Please fit some of the concepts of this chapter to this situation, and calculate the probability.