Chapt 4b - 4.4 Geometric Distribution 121 = P [ ( X = n + k...

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Unformatted text preview: 4.4 Geometric Distribution 121 = P [ ( X = n + k ) ( X > n ) ] P [ X > n ] = P [ X = n + k ] P [ X > n ] = p ( 1- p ) n + k- 1 ( 1- p ) n = p ( 1- p ) k- 1 = p X ( k ) k = 1 , 2 ,... where the fourth equality follows from the fact that the event { ( X = n + k ) ( X > n ) } is equal to the event { X = n + k } and the fifth equality follows from our earlier result that P [ X > n ] = ( 1- p ) n . The above result shows that the conditional probability that the number of trials remaining until the first success, given that no success occurred in the first n trials, has the same PMF as X had originally. This property is called the forget- fulness or memorylessness property of the geometric distribution. It means that the distribution forgets the past by not remembering how long the sequence has lasted if no success has already occurred. Thus, each time we want to know the number of trials until the first success, given that the first success has not yet occurred, the process starts from the beginning. Example 4.8 Tony is tossing balls randomly into 50 boxes, and his goal is to stop when he gets the first ball into the eighth box. Given that he has tossed 20 balls without getting a ball into the eighth box, what is the expected number of additional tosses he needs to get a ball into the eighth box? Solution With respect to the eighth box, each toss is a Bernoulli trial with proba- bility of success p = 1 / 50. Let X be the random variable that denotes the number of tosses required to get a ball into the eighth box. Then X has a geometric distri- bution. Let K denote the number of additional tosses required to get a ball into the eighth box, given that no ball is in the box after 20 tosses. Then, because of the forgetfulness property of the geometric distribution, K has the same distribution as X . Thus E [ K ] = 1 p = 50 trianglesolid 122 Chapter 4 Special Probability Distributions 4.5 Pascal (or Negative Binomial) Distribution The Pascal random variable is an extension of the geometric random variable. It describes the number of trials until the k th success, which is why it is some- times called the k th-order interarrival time for a Bernoulli process. The Pascal distribution is also called the negative binomial distribution . Let the k th success in a Bernoulli sequence of trials occur at the n th trial. This means that k- 1 successes occurred during the past n- 1 trials. From our knowledge of the binomial distribution, we know that the probability of this event is p X ( n- 1 ) ( k- 1 ) = parenleftbigg n- 1 k- 1 parenrightbigg p k- 1 ( 1- p ) n- k where X ( n- 1 ) is the binomial random variable with parameters ( n- 1 , p ) . The next trial independently results in a success with probability p . Thus, if we define the Bernoulli random variable as X 1 whose PMF we defined earlier as p X 1 ( x ) = p x ( 1- p ) 1- x , where x = 0 or 1, the PMF of the k th-order Pascal random variable, X k , is given by p X k ( n ) = P [{ X ( n...
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Chapt 4b - 4.4 Geometric Distribution 121 = P [ ( X = n + k...

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