AP Statistics Curriculum 2007 Distrib Binomial
General Advance-Placement (AP) Statistics Curriculum - Bernoulli and Binomial Random Variables and Experiments
Bernoulli process
A Bernoulli trial is an experiment whose dichotomous outcomes are random (e.g., "success" vs. "failure", "head" vs. "tail", +/-, "yes" vs. "no", etc.) Most common notations of the outcomes of a Bernoulli process are success and failure, even though these outcome labels should not be construed literally.
- Examples of Bernoulli trials
- A Coin Toss. We can obverse H="heads", conventionally denoted success, or T="tails" denoted as failure. A fair coin has the probability of success 0.5 by definition.
- Rolling a Die: Where the outcome space is binarized to "success"={6} and "failure" = {1, 2, 3, 4, 5}.
- Polls: Choosing a voter at random to ascertain whether that voter will vote "yes" in an upcoming referendum.
- The Bernoulli random variable (RV): Mathematically, a Bernoulli trial is modeled by a random variable \(X(outcome) = \begin{cases}0,& s = \texttt{success},\\ 1,& s = \texttt{failure}.\end{cases}\) If p=P(success), then the expected value of X, E[X]=p and the standard deviation of X, SD[X] is \(\sqrt{p(1-p)}\).
- A Bernoulli process consists of repeatedly performing independent but identical Bernoulli trials.
Binomial Random Variables
Suppose we conduct an experiment observing an n-trial (fixed) Bernoulli process. If we are interested in the RV X={Number of successes in the n trials}, then X is called a Binomial RV and its distribution is called Binomial Distribution.
Examples
- Roll a standard die ten times. Let X be the number of times {6} turned up. The distribution of the random variable X is a binomial distribution with n = 10 (number of trials) and p = 1/6 (probability of "success={6}). The distribution of X may be explicitely written as (P(X=x) are rounded of, you can compute these exactly by going to [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distributions and selecting Binomial):
| x | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| P(X=x) | 0.162 | 0.323 | 0.291 | 0.155 | 0.0543 | 0.013 | 0.0022 | 0.00025 | 0.000019 | 8.269e-7 | 1.654e-8 |
- Suppose 10% of the human population carries the green-eye alial. If we choose 1,000 people randomly and let the RV X be the number of green-eyed people in the sample. Then the distribution of X is binomial distribution with n = 1,000 and p = 0.1 (denoted as \(X \sim B(1,000, 0.1)\).
Approach
Models & strategies for solving the problem, data understanding & inference.
- TBD
Model Validation
Checking/affirming underlying assumptions.
- TBD
Computational Resources: Internet-based SOCR Tools
- TBD
Examples
Computer simulations and real observed data.
- TBD
Hands-on activities
Step-by-step practice problems.
- TBD
References
- TBD
- SOCR Home page: http://www.socr.ucla.edu
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