# AP Statistics Curriculum 2007 Distrib Multinomial

(Difference between revisions)
 Revision as of 23:30, 4 March 2008 (view source)IvoDinov (Talk | contribs)← Older edit Revision as of 23:50, 4 March 2008 (view source)IvoDinov (Talk | contribs) Newer edit → Line 3: Line 3: The multinomial experiments (and multinomial distribtuions) directly extend the their [[AP_Statistics_Curriculum_2007_Distrib_Binomial |bi-nomial counterparts]]. The multinomial experiments (and multinomial distribtuions) directly extend the their [[AP_Statistics_Curriculum_2007_Distrib_Binomial |bi-nomial counterparts]]. - * Examples of Multinomial experiments + ===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: $1\leq value\leq 6$. + A multinomial experiment is an experiment that has the following properties: + * The experiment consists of '''k repeated trials'''. + * Each trial has a '''discrete''' number of possible outcomes. + * On any given trial, the probability that a particular outcome will occur is '''constant'''. + * The trials are '''independent'''; that is, the outcome on one trial does not affect the outcome on other trials. - * The Multinomial random variable (RV): Mathematically, a (k) multinomial trial is modeled by a random variable + ====Examples of Multinomial experiments==== -
$X(outcome) = \begin{cases}x_o,\\ + * Suppose we have an urn containing 9 marbles. Two are red, three are green, and four are blue (2+3+4=9). We randomly select 5 marbles from the urn, ''with replacement''. What is the probability (''P(A)'') of the event ''A={selecting 2 green marbles and 3 blue marbles}''? - x_1,\\ + - \cdots,\\ + - x_k.\end{cases}$
+ - If $p_i=P(X=x_i)$, then: + *To solve this problem, we apply the multinomial formula. We know the following: - : ''expected value'' of X, $E[X]=\sum_{i=1}^k{x_i\times p_i}$. + ** The experiment consists of 5 trials, so k = 5. - : standard deviation of X, $SD[X]=\sqrt{\sum_{i=1}^k{(x_i-E[X])^2\times p_i}}$. + ** The 5 trials produce 0 red, 2 green marbles, and 3 blue marbles; so $r_1=r_{red} = 0$, $r_2=r_{green} = 2$, and $r_3=r_{blue} = 3$. + ** For any particular trial, the probability of drawing a red, green, or blue marble is 2/9, 3/9, and 5/9, respectively. Hence, $p_1=p_{red} = 2/9$, $p_2=p_{green} = 1/3$, and $p_3=p_{blue} = 5/9$. + + Plugging these values into the multinomial formula we get the probability of the event of interest to be: + + : $P(A) = {5\choose 0, 2, 3}p_1^{r_1}p_2^{r_2}p_3^{r_3}$ + + : $P(A) = {5! \over 0!\times 2! \times 3! }\times (2/9)^0 \times (1/3)^2\times (5/9)^3=0.19052.$ + + Thus, if we draw 5 marbles with replacement from the urn, the probability of drawing no red , 2 green, and 3 blue marbles is ''0.19052''. ===Synergies between Binomial and Multinomial processes/probabilities/coefficients=== ===Synergies between Binomial and Multinomial processes/probabilities/coefficients===

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

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

### Multinomial experiments

A multinomial experiment is an experiment that has the following properties:

• The experiment consists of k repeated trials.
• Each trial has a discrete number of possible outcomes.
• On any given trial, the probability that a particular outcome will occur is constant.
• The trials are independent; that is, the outcome on one trial does not affect the outcome on other trials.

#### Examples of Multinomial experiments

• Suppose we have an urn containing 9 marbles. Two are red, three are green, and four are blue (2+3+4=9). We randomly select 5 marbles from the urn, with replacement. What is the probability (P(A)) of the event A={selecting 2 green marbles and 3 blue marbles}?
• To solve this problem, we apply the multinomial formula. We know the following:
• The experiment consists of 5 trials, so k = 5.
• The 5 trials produce 0 red, 2 green marbles, and 3 blue marbles; so r1 = rred = 0, r2 = rgreen = 2, and r3 = rblue = 3.
• For any particular trial, the probability of drawing a red, green, or blue marble is 2/9, 3/9, and 5/9, respectively. Hence, p1 = pred = 2 / 9, p2 = pgreen = 1 / 3, and p3 = pblue = 5 / 9.

Plugging these values into the multinomial formula we get the probability of the event of interest to be:

$P(A) = {5\choose 0, 2, 3}p_1^{r_1}p_2^{r_2}p_3^{r_3}$
$P(A) = {5! \over 0!\times 2! \times 3! }\times (2/9)^0 \times (1/3)^2\times (5/9)^3=0.19052.$

Thus, if we draw 5 marbles with replacement from the urn, the probability of drawing no red , 2 green, and 3 blue marbles is 0.19052.

### 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) = ?