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Some Common Discrete Random Variables

Some Common Discrete Random Variables. Binomial Random Variables. Binomial experiment. A sequence of n trials ( called Bernoulli trials ), each of which results in either a “success” or a “failure”.

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Some Common Discrete Random Variables

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  1. Some Common Discrete Random Variables

  2. Binomial Random Variables

  3. Binomial experiment • A sequence ofn trials(called Bernoulli trials), each of which results in either a “success” or a “failure”. • The trials are independent and so theprobability of success, p, remains the same for each trial. • Define a random variable Y as the number of successes observed during the n trials. • What is the probability p(y), for y = 0, 1, …, n ? • How many successes may we expect? E(Y) = ?

  4. Returning Students • Suppose the retention rate for a school indicates the probability a freshman returns for their sophmore year is 0.65. Among 12 randomly selected freshman, what is the probability 8 of them return to school next year? Each student either returns or doesn’t. Think of each selected student as a trial, so n = 12. If we consider “student returns” to be a success, then p = 0.65.

  5. 12 trials, 8 successes • To find the probability of this event, consider the probability for just one sample point in the event. • For example, the probability the first 8 students return and the last 4 don’t. • Since independent, we just multiply the probabilities:

  6. 12 trials, 8 successes • For the probability of this event, we sum the probabilities for each sample point in the event. • How many sample points are in this event? • How many ways can 8 successes and 4 failures occur? • Each of these sample points has the same probability. • Hence, summing these probabilities yields

  7. Binomial Probability Function • A random variable has a binomial distribution with parameters n and p if its probability function is given by

  8. Rats! • In a research study, rats are injected with a drug. The probability that a rat will die from the drug before the experiment is over is 0.16. Ten rats are injected with the drug. What is the probability that at least 8 will survive? Would you be surprised if at least 5 died during the experiment?

  9. Quality Control • For parts machined by a particular lathe, on average, 95% of the parts are within the acceptable tolerance. • If 20 parts are checked, what is the probability that at least 18 are acceptable? • If 20 parts are checked, what is the probability that at most 18 are acceptable?

  10. Binomial Theorem • As we saw in our Discrete class, the Binomial Theorem allows us to expand • As a result, summing the binomial probabilities, where q = 1- p is the probability of a failure,

  11. Mean and Variance • If Y is a binomial random variable with parameters n and p, the expected value and variance for Y are given by

  12. Rats! • In a research study, rats are injected with a drug. The probability that a rat will die from the drug before the experiment is over is 0.16. Ten rats are injected with the drug. • How many of the rats are expected to survive? • Find the variance for the number of survivors.

  13. Geometric Random Variables

  14. Your 1st Success • Similar to the binomial experiment, we consider: • A sequence ofindependent Bernoulli trials. • The probability of “success” equals p on each trial. • Define a random variable Y as the number of the trial on which the 1st success occurs. (Stop the trials after the first success occurs.) • What is the probability p(y), for y = 1,2, … ? • On which trial is the first success expected?

  15. (S) S (F, S) S F (F, F, S) S F (F, F, F, S) S F …. S = success • Consider the values of Y:y = 1: (S)y = 2: (F, S)y = 3: (F, F, S)y = 4: (F, F, F, S)and so on… p(1) = pp(2) = (q)( p)p(3) = (q2)( p)p(4) = (q3)( p)

  16. Geometric Probability Function • A random variable has a geometric distribution with parameter p if its probability function is given by

  17. (D) D (G, D) D G (G, G, D) D G G Success? • Of course, you need to be clear on what you consider a “success”. • For example, the 1st success might mean finding the 1st defective item!

  18. Geometric Mean, Variance • If Y is a geometric random variable with parameter p the expected value and variance for Y are given by

  19. At least ‘a’ trials? (#3.55) • For a geometric random variable and a > 0,show P(Y > a) = qa • Consider P(Y > a) = 1 – P(Y < a) = 1 – p(1 + q + q2 + …+ qa-1) = qa , based on the sum of a geometric series

  20. “Memoryless Property” • For the geometric distribution P(Y > a + b | Y > a ) = qb = P(Y > b) • “at least 5 more trials?” We note P(Y > 7 | Y > 2 ) = q5 = P(Y > 5). That is, “knowing the first two trials were failures, the probability a success won’t occur on the next 5 trials” is identical to…“just starting the trials and a success won’t occur on the first 5 trials”

  21. Negative Binomial Distribution • Again, considering a independent Bernoulli trials with probability of “success” p on each trial… • Instead of watching for the 1st success, let Y be the number of the trial on which the rth success occurs. (Stop the trials after the rth success occurs.) • For a given value r, the probability p(y) is

  22. Negative Binomial • To determine the probability the 4th success occurs on the 7th trial, we compute • Note this is actually just the binomial probability of 3 successes during the first 6 trials, followed by one more success: “a success on 4th last trial”

  23. Negative Binomial • For the negative binomial distribution, we have • For example, if a success occurs 10% of the time (i.e., p = 0.1), then to find the 4th success, we expect to require 40 trials on average. Intuitively, wouldn’t you expect 40 trials?

  24. Poisson Random Variables

  25. Number of occurrences • Let Y represent the number of occurrences of an event in an interval of size s. • Here we may be referring to an interval of time, distance, space, etc. • For example, we may be interested in the number of customers Y arriving during a given time interval. • We call Y a Poisson random variable.

  26. Poisson R. V. • A random variable has a Poisson distribution with parameter l if its probability function is given by where y = 0, 1, 2, … We’ll see that l is the “average rate” at which the events occur. That is, E(Y) = l .

  27. Queries • If the number of database queries processed by a computer in a time interval is a Poisson random variable with an average of 6 queries per minute, find the probability that 4 queries occur in a one minute interval.

  28. Fewer Queries • As before, for the Poisson random variable with an average of 6 queries per minute… • find the probability there are less than 6 queries in a one minute interval:

  29. Some PoissonVariables • Number of incoming telephone calls to a switchboard within a given time interval; • Number of errors (incorrect bits) received by a modem during a given time interval; • Number of chocolate chips in one of Dr. Vestal’s chocolate chip cookies; • Number of claims processed by a particular insurance company on a single day; • Number of white blood cells in a drop of blood; • Number of dead deer along a mile of highway.

  30. Poisson mean, variance • If Y is a Poisson random variable with parameter l, the expected value and variance for Y are given by

  31. Hypergeometric Random Variables

  32. Sampling without replacement • When sampling with replacement, each trial remains independent. For example,… • If balls are replaced, P(red ball on 2nd draw) =P(red ball on 2nd draw | first ball was red). • If balls not replaced, then given the first ball is red, there is less chance of a red ball on the 2nd draw. Though for a large population of balls, the effect may be minimal.

  33. n trials, y red balls • Suppose there are r red balls, and N – r other balls. • Consider Y, the number of red balls in n selections,where now the trials may be dependent.(for sampling without replacement, when sample size is significant relative to the population) • The probability y of the n selected balls are red is

  34. Hypergeometric R. V. • A random variable has a hypergeometric distribution with parameters N, n, and r if its probability function is given by where 0 <y< min( n, r ).

  35. Hypergeometric mean, variance • If Y is a hypergeometric random variable with parameter p the expected value and variance for Y are given by

  36. Sample of 20 Suppose among a supply of 5000 parts produced during a given week, there are 100 that don’t meet the required quality standard. Twenty of the parts are randomly selected and checked to see if they meet the standard. Let Y be the number in the sample that don’t meet the standard.a). Compute the probability exactly 2 of the sampled parts fail to meet the quality standard.b). Determine the mean, E(Y).

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