AP Statistics Curriculum 2007 Distrib Multinomial

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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




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