AP Statistics Curriculum 2007 Distrib Multinomial

From Socr

Revision as of 23:32, 4 March 2008 by IvoDinov (Talk | contribs)
Jump to: navigation, search

Contents

General Advance-Placement (AP) Statistics Curriculum - Multinomial Random Variables and Experiments

The multinomial experiments (and multinomial distribtuions) directly extend the their bi-nomial counterparts.

  • Examples of Multinomial experiments
    • Rolling a hexagonal Die 5 times: Where the outcome space is the colection of 5-tuples, where each element is a value such that: 1\leq value\leq 6.
  • The Multinomial random variable (RV): Mathematically, a (k) multinomial trial is modeled by a random variable
X(outcome) = \begin{cases}x_o,\\ x_1,\\ \cdots,\\ x_k.\end{cases}

If pi = P(X = xi), then:

expected value of X, E[X]=\sum_{i=1}^k{x_i\times p_i}.
standard deviation of X, SD[X]=\sqrt{\sum_{i=1}^k{(x_i-E[X])^2\times p_i}}.

Synergies between Binomial and Multinomial processes/probabilities/coefficients

  • The Binomial vs. Multinomial Coefficients
{n\choose i}=\frac{n!}{k!(n-k)!}
{n\choose i_1,i_2,\cdots, i_k}= \frac{n!}{i_1! i_2! \cdots i_k!}
  • The Binomial vs. Multinomial Formulas
(a+b)^n = \sum_{i=1}^n{{n\choose i}a^1 \times b^{n-i}}
(a_1+a_2+\cdots +a_k)^n = \sum_{i_1+i_2\cdots +i_k=n}^n{ {n\choose i_1,i_2,\cdots, i_k} a_1^{i_1} \times a_2^{i_2} \times \cdots \times a_k^{i_k}}
  • The Binomial vs. Multinomial Probabilities
p=P(X=r)={n\choose i}p^r(1-p)^{n-r}, \forall 0\leq r \leq n
p=P(X_1=r_1 \cap X_1=r_1 \cap \cdots \cap X_k=r_k | r_1+r_2+\cdots+r_k=n)={n\choose i_1,i_2,\cdots, i_k}p_1^{r_1}p_2^{r_2}\cdots p_k^{r_k}, \forall r_1+r_2+\cdots+r_k=n


Example

Suppose we study N independent trials with results falling in one of k possible categories labeled 1,2,cdots,k. Let pi be the probability of a trial resulting in the ith category, where p_1+p_2+\cdots++p_k =1. Let Ni be the number of trials resulting in the ith category, where N_1+N_2+\cdots++N_k = N.

For instance, suppose we have 9 people arriving at a meeting according to the following information:

P(by Air) = 0.4, P(by Bus) = 0.2, P(by Automobile) = 0.3, P(by Train) = 0.1
  • Compute the following probabilities
P(3 by Air, 3 by Bus, 1 by Auto, 2 by Train) = ?
P(2 by air) = ?


SOCR Multinomial Examples

References

Translate this page:

(default)

Deutsch

Español

Français

Italiano

Português

日本語

България

الامارات العربية المتحدة

Suomi

इस भाषा में

Norge

한국어

中文

繁体中文

Русский

Nederlands

Ελληνικά

Hrvatska

Česká republika

Danmark

Polska

România

Sverige

Retrieved from "http://wiki.stat.ucla.edu/socr/index.php/AP_Statistics_Curriculum_2007_Distrib_Multinomial"
Personal tools