Geometric Distribution The geometric distribution computes the probability

Geometric Distribution

The geometric distribution computes the probability that the first success of a Bernoulli trial comes on the kth trial. For example, suppose we want to roll a four on a six-sided die. We will roll until we get a four, then we stop. The random variable is the number of rolls until we get a four. The probability of getting a four on the first roll is P(1)=1/6. The probability of getting a four for the first time on the second roll is P(2)=(5/6)(1/6)=5/36.

The probability of getting the first four on the third roll is P(3)=(5/6)(1/6)=25/216. In general, we see that all but the last roll are not fours and so have a probability of 5/6 and the last roll will have a probability of 1/6. So, P(X=k)=(5/6)(k-1)(1/6) Generalizing to other problems, it is not so difficult to see that as there are k-1 failures followed by one success.

We can work many problems on the TI-83, although the formula is so simple that you may just want to work them directly. Problem: Compute the probability that the first four occurs on the third roll of the die. On the calculator, press <2 nd> <DISTR> <D: geometpdf(> then type in 1/6, 3). (You can convert to a fraction if you like with <MATH> <1: Frac> <ENTER>. ) As you see, we get the same answer as we got earlier.

We can use a sequence to get several probabilities at once. Problem: Compute the probabilities that the first four occurs on the first through fifth rolls of a fair die. This example will be worked on the homescreen so that we can see fractions. Most often I work in lists, as we did with the binomial distribution. Press: <2 nd> <DISTR> <D: geometpdf(> then type 1/6, {1, 2, 3, 4, 5}) <ENTER>. Now press: <MATH> <1: Frac> <ENTER> Hint: The { symbol uses the 2 nd function, then the ( key. Use the toggle arrows to scroll to see the other answers. {1/6, 5/6, 25/216, 125/1296, 625/7776}

Problem: Find the probability of getting a four in the first three trials, i. e. , P(X 3). In this case we use the cumulative density function geometcdf. On the TI-83, press: <2 nd> <DISTR> <E: geometcdf(> then type 1/6, 3) As you see, we find the probability is 91/216. We can also make relative frequency histograms for the pdf and the cdf.

Problem: Construct relative frequency histograms for the first four on rolls of a six-sided die. Show X=1 through X=7. Enter 1 through 7 in L 1. Let L 2 = geometpdf(1/6, L 1). Set a window as shown, and set up the STAT PLOT. Trace to see the area of each column in the histogram.

Last Problem: Plot the histogram for the cumulative density function for the same experiment. Show X=1 through X=7. Enter 1 through 7 in L 1. Let L 2 = geometcdf(1/6, L 1). Set a window as shown, and set up the STAT PLOT. Trace to see the area of each column in the histogram.

The end