AP Statistics Curriculum 2007 Distrib Dists
Contents
General Advance-Placement (AP) Statistics Curriculum - Geometric, HyperGeometric, Negative Binomial Random Variables and Experiments
Geometric
- Definition: The Geometric Distribution is the probability distribution of the number X of Bernoulli trials needed to get one success, supported on the set {1, 2, 3, ...}.
- Mass Function: If the probability of successes on each trial is P(success)=p, then the probability that x trials are needed to get one success is \(P(X = x) = (1 - p)^{x-1} \times p\), for x = 1, 2, 3, 4,....
- Expectation: The Expected Value of a geometrically distributed random variable X is \({1\over p}.\)
- Variance: The Variance is \({1-p\over p^2}.\)
- Example: See this SOCR Geometric distribution activity.
HyperGeometric
The hypergeometric distribution is a discrete probability distribution that describes the number of successes in a sequence of n draws from a finite population without replacement. An experimental design for using Hypergeometric distribution is illustrated in this table:
Type | Drawn | Not-Drawn | Total |
Defective | k | m-k | m |
Non-Defective | n-k | N+k-n-m | N-m |
Total | n | N-n | N |
- Explanation: Suppose there is a shipment of N objects in which m are defective. The Hypergeometric Distribution describes the probability that in a sample of n distinctive objects drawn from the shipment exactly k objects are defective.
- Mass function: The random variable X follows the Hypergeometric Distribution with parameters N, m and n, then the probability of getting exactly k successes is given by
\[ P(X=k) = {{{m \choose k} {{N-m} \choose {n-k}}}\over {N \choose n}}.\]
This formula for the Hypergeometric Mass Function may be interpreted as follows: There are \({{N}\choose{n}}\) possible samples (without replacement). There are \({{m}\choose{k}}\) ways to obtain k defective objects and there are \({{N-m}\choose{n-k}}\) ways to fill out the rest of the sample with non-defective objects.
Examples
- SOCR Activity: The SOCR Ball and Urn Experiment provides a hands-on demonstration of the utilization of Hypergeometric distribution in practice. This activity consists of selecting n balls at random from an urn with N balls, R of which are red and the other N - R green. The number of red balls Y in the sample is recorded on each update. The distribution and moments of Y are shown in blue in the distribution graph and are recorded in the distribution table. On each update, the empirical density and moments of Y are shown in red in the distribution graph and are recorded in the distribution table. Either of two sampling models can be selected with the list box: with replacement and without replacement. The parameters N, R, and n can be varied with scroll bars.
- A lake contains 1,000 fish; 100 are randomly caught and tagged. Suppose that later we catch 20 fish. Use SOCR Hypergeometric Distribution to:
- Compute the probability mass function of the number of tagged fish in the sample of 20.
- Compute the expected value and the variance of the number of tagged fish in this sample.
- Compute the probability that this random sample contains more than 3 tagged fish.
- You can also see a manual calculation example using the hypergeometric distribution here.
Negative Binomial
- Definition: The family of Negative Binomial Distributions is a two-parameter family; p and r with 0 < p < 1 and r > 0.
- Mass Function: The probability mass function of a Negative Binomial random variable (X~NegBin(r, p))is give by:
\[ P(X=k) = {k+r-1 \choose k}\cdot p^r \cdot (1-p)^k \!\], for k = 0,1,2,....
Application
Suppose Jane is promoting and fund-raising for a presidential candidate. She wants to visit all 50 states and she's pledged to get all electoral votes of 6 states before she and the candidate she represents are satisfied. In every state, there is a 30% chance that Jane will be able to secure all electoral votes and 70% chance that she'll fail.
- What's the probability mass function that the last 6^{th} state where she succeeds to secure all electoral votes happens to be the at the n^{th} state she campaigns in?
NegBin(r, p) distribution describes the probability of k failures and r successes in n=k+r Bernoulli(p) trials with success on the last trial. Looking to secure the electoral votes for 6 states means Jane needs to get 6 successes before she (and her candidate) is happy. The number of trials (i.e., states visited) needed is n=k+6. The random variable we are interested in is X={number of states visited to achieve 6 successes (secure all electoral votes within these states)}. So, k = n-6, and \(X\sim NegBin(6, 0.3)\). Thus, for \(n \geq 6\), the mass function (giving the probabilities that Jane will visit n states before her ultimate success is:
\[ P(X=n) = {(n-6) + 6 - 1 \choose 6-1} \; 0.3^6 \; 0.7^{n-6} = {n-1 \choose 5} \; \frac{3^6 \; 7^{n-6} \; }{10^n} \]
- What's the probability that Jane finishes her campaign in the 10^{th} state?
\[ P(X=10) = 0.022054\]
- What's the probability that Jane finishes campaigning on or before reaching the 8^{th} state?
\[ P(X\leq 8) = 0.011292\]
- Suppose the success of getting all electoral votes within a state is reduced to only 10%, then X~NegBin(r=6, p=0.1). Notice that the shape and domain the Negative-Binomial distribution significantly chance now (see image below)! What's the probability that Jane covers all 50 states but fails to get all electoral votes in any 6 states (as she had hoped for)?
\[ P(X\geq 50) = 0.632391\]
- SOCR Activity: If you want to see an interactive Negative-Binomial Graphical calculator you can go to this applet (select Negative Binomial) and see this activity.
References
- SOCR Home page: http://www.socr.ucla.edu
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