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Handout 9 Supplements material in Section 3.4 of L-G Equality of RV’s Consider and as two random variables. There are lots of ways one might think of these as the same random variable. One way might be if both and are defined on the same sample space , and they are exactly the same function. They are then called (identically) equal random variables. A second (and more common) type of equality is if the distribution functions are the same. This is best done in an example. Let be a random variable which records the number of heads in 2 tosses of a coin. can take on the values with probabilities . Now consider another random variable, , which is determined by the throw of a sided dice. is defined as if the dice shows 1 if the dice shows 2 or 3 if the dice shows 3 Now, can take on the values with probabilities . Clearly, and are not the same random variable. They are not even defined on the same sample space. Equally clearly, once we abstract from the sample space to the random variable, both and have the same behavior. and are considered to be identically distributed random variables and we write this as Discrete Random Variables We have already seen a couple of discrete random variables in Chapter 2 without using the words “Random Variable”. The Binomial Random Variable corresponds to the number of successes out of independent repetitions of an experiment, where the probability of success in a single experiment is . The probability function of the Binomial random variable is for . The cdf of the Binomial rv cannot be written in a closed-form expression. The Geometric Random Variable corresponds to the number of repetitions of the experiment it takes to achieve the first success. The probability function of the Geometric rv is 1 36-217: Probability Theory and Random Processes for Fall 1997 . The cdf of the Geometric rv can be written in a closed form expression and it is A very important property of the Geometric random variable is the memoryless property: for all . This states that if a success has not occurred in the first trials, then the probability of having to repeat the experiment more times in order to get a success is the same as the probability of initially performing the experiment times in order to get a success. The geometric random variable arises in applications where one is interested in time until the occurrence of an event or between two events. Examples on the Geometric Distribution Example 9.1 An oil prospector will drill a succession of holes in a given area to find a productive well. The probability that he is successful on a given trial is 0.2. (a) What is the probability that the third hole drilled is the first that yields a productive well? (b) If the prospector can only afford o drill at most ten wells, what is the probability that he fails to find a productive well? Solution. Let’s define the random variable which denotes the number of holes needed to be drilled in order to get the first productive well. Then will follow a Geometric Distribution with parameter . (a) the third hole is the first success (b) first 10 holes are nonproductive 2 36-217: Probability Theory and Random Processes Fall 1997 The Negative Binomial Random Variable The Negative Binomial Distribution A random variable with a negative binomial distribution originates from a context that is very similar to the one that leads to the geometric distribution. Again we focus on independent and identical trials each of which results on one of two outcomes, “success” or “failure”. The probability of success is and stays the same from trial to trial. The Geometric distribution handles the case where we are interested in the number of trials until the first success occurs. What if we are interested in the number of the trial on which the second, third of fourth success occurs? The distribution that applies to the random variable , equal to the number of the trial on which the th success occurs ( ) is the Negative Binomial Distribution. Let’s, now, find the probability distribution of . the th trial is the one in which the th success occurs successes in the first trials) and a success on the th trial where Example 9.2 A geological study indicates that an exploratory oil well drilled in a particular region should strike oil with probability 0.2. Find the probability that the third oil strike comes on the fifth well drilled. Solution. Let denote the number of the trial on which the third oil strike occurs. Then it is reasonable to assume that has negative binomial distribution with parameters and . Thus 3 36-217: Probability Theory and Random Processes Fall 1997 The Poisson Distribution Consider an experiment where you set up a Geiger counter and count the number of clicks (say particle emissions) in a fixed period of time. There is clearly no limit to the number of clicks (for practical purposes there is a limit, but let’s ignore that), so the range of the number of clicks is . Other examples include the number of packets on an Ethernet in some time period, the number of cars passing by Forbes and Morewood for a given length of time etc. The random variable that denotes such counts as above is said to be the Poisson random variable. The probability distribution function of the Poisson random variable is given by the following expression: The parameter will turn out to be important–it is actually the mean of the distribution. Example 9.3 There are two entrances to a parking lot. Cars arrive at entrance I according to a Poisson distribution at an average of three an hour, and a entrance II according to a Poisson distribution at an average of four per hour. What is the probability that three cars arrive at the parking lot in a given hour? (Assume that the numbers of cars arriving at the two entrances are independent) Solution. Let be the number of cars entering the tunnel in a given 2-minute period. Poisson distribution with . Then has a There are trials now, and let the number of times (out of 10) that the event occurs. Then follows a Binomial distribution with parameters and . Thus the probability of interest becomes: Next time: The Poisson rv (continued) and Continuous rvs 4