MULTINOMIAL PROBABILITY DISTRIBUTION

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Lorraine Lawrence
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1 MTH/STA 56 MULTINOMIAL PROBABILITY DISTRIBUTION The multinomial probability distribution is an extension of the binomial probability distribution when the identical trial in the experiment has more than two possible outcomes As a result, the multinomial experiment can be easily extended from the binomial experiment as follows The multinomial probability distribution is an extension of the binomial probability distribution when the identical trial in the experiment has more than two possible outcomes As a result, the multinomial experiment can be easily extended from the binomial experiment as follows De nition A multinomial experiment is one that possesses the following properties: The experiment consists of n repeated identical trials Each trial results in an outcomes that may be classi ed as one of k classes or cells 3 The probability that the outcome of a single trial will fall into a particular cell, say, cell j, is denoted by p j (j = ; ; ; k) and remains constant from trial to trial Note that p + p + + p k = 4 The repeated trials are independent 5 The random variable of interest are Y, Y,, Y k, where Y j (j = ; ; ; k) is the number of trials for which the outcome falls into cell j Note that Y + Y + + Y k = n Example The following are typical multinomial experiments: Toss a balanced die ten times and observe the number of dot j for j = ; ; ; 6 According to the theory of genetics, a certain cross of guinea pigs will result in red, black, and white o springs in the ratio 8 : 4 : 4 Among 6 o springs the random variables of interest are the number of reds, blacks, and whites 3 Suppose that the probabilities are 0:4, 0:, 0:3, and 0:, respectively, that a delegate to a certain convention arrived by air, bus, automobile, or train Among 0 delegates the random variables of interest are the number of delegates arrived by air, bus, automobile, or train, respectively Theorem (multinomial probability distribution) If a given trial can result in the k outcomes E, E,, E k with probabilities p, p,, p k, then the joint probability distribution of the random variables Y, Y,, Y k, representing the number of occurrences for E, E,, E k in n independent trials, is given by p (y ; y ; ; y k ) = n! y!y! y k! py p y p y k k where y + y + + y k = n and p + p + + p k =

2 Proof Since the trials are independent, any special order yielding y outcomes for E, y outcomes for E,, y k outcomes for E k will occur with probability p y p y p y k k The total number of orders yielding similar outcomes for the n trials is equal to the number of partitions of n items into k groups with y in the rst group, y in the second group,, y k in the kth group this can be done in n! y!y! y k! ways Since all partitions are mutually exclusive and occur with equal probability, we obtain the multinomial [probability distribution by multiplying the probability for a speci ed order by the total number of partitions Example Suppose that crossing two particular hybrid petunias leads to plants with one of four di erent types of owers, red-serrated edge, red-smooth edge, white-serrated edge, and white-smooth edge Also, assume that the separate plants are independent and p = 0:, p = 0:3, p 3 = 0:, and p 4 = 0:4 If 0 plants were produced from the cross, it follows from the multinomial probability distribution that 0! P fy = ; Y = 4; Y 3 = 3; Y 4 = g =!4!3!! (0:) (0:3) 4 (0:) 3 (0:4) = 0:003659: Example 3 Suppose that a pair of balanced dice is tossed 6 times, what is the probability of observing a total of 7 or twice, a matching pair once, and any other combinations three times? Solution Let us rst list the following possible outcomes for each toss: E : a total of 7 or occurs E : a matching pair occurs E 3 : neither a matching pair nor a total of 7 or occurs The corresponding probabilities for a given trial are p = 8 36 = 9 p = 6 36 = 6 p 3 = 36 = 8 : These values remain constant for all 6 trials Using the multinomial probability distribution with Y =, Y =, Y 3 = 3, the desired probability is given by p (; ; 3) = 6!!!3! = 0:7: Note Similar to the binomial case, the multinomial expansion is given by (a + a + + a k ) n = X X X n! n!n! n k! an a n a n k k : n +n ++n k =n

3 Since p + p + + p k =, we conclude that X X X p (y ; y ; ; y k ) = X X X n! y!y! y k! py y +y ++y k =n y +y ++y k =n p y = (p + p + + p k ) n = : p y k k Theorem If random variables Y, Y,, Y k have a multinomial probability distribution with parameters p, p,, p k, respectively, then, for j = ; ; ; k, E (Y j ) = np j and V ar (Y j ) = np j q j ; where q j = p j ; and s = ; ; ; k and t = ; ; ; k, where q t = p t Cov (Y s ; Y t ) = np s q t if s 6= t; Proof To nd the expected value and variance of Y j, let us rst obtain the marginal probability distribution for Y j Recall that Y j is the number of trials that are classi ed into cell j Combine all of the cells, excluding cell j, into a single large cell Now we have only two mutually exclusive cells ( cell j and not cell j ) with probabilities p j and p j, respectively Then every trial will fall into either cell j or not cell j Thus, Y j has a binomial probability distribution with parameters n and p j Consequently, where q j = p j E (Y j ) = np j and V ar (Y j ) = np j q j ; To nd the covariance of Y s and Y t, we need to de ne two sets of Bernoulli random variables, one for cell s and the other for cell, as follows: For cell s, we de ne, for each trial i (i = ; ; ; n), U i = if the outcome of trial i falls into cell s 0 otherwise For example, U 3 = indicates that the outcome of the third trial falls into cell s while U 5 = 0 implies that the outcome of the fth trial do not fall into cell s We must note that U i = or 0 depending upon whether the outcome of trial i falls into cell s, and Y s, being the number of times cell s is observe, can be interpreted as simply the sum of a series of zeros and ones in which a one occurs in the sum every time we observe an item from cell s, and a zero occurs every time we observe any other cells Thus, Y s = U i : Likewise, for cell t, we de ne, for each trial i (i = ; ; ; n), if the outcome of trial i falls into cell t W i = 0 otherwise 3

4 For instance, W = indicates that the outcome of the second trial falls into cell t while W 4 = 0 implies that the outcome of the fourth trial do not fall into cell t Then W i = or 0 depending upon whether the outcome of trial i falls into cell t Note that Y t (the number of times cell t is observed) can be interpreted as simply the sum of a series of zeros and ones in which a one occurs in the sum every time we observe an item from cell t, and a zero occurs every time we observe any other cells Thus, Y t = W i : The subscripts s, t, and i could be very confused Bear in your mind that subscripts s and t refer to cell s and t, respectively, while subscript i refers to trial i The following table should be very helpful in clarifying this confusion: Cell Trial s t k U W U W i U i W i n U n W n Sum Y s = U i Y t = W i By de nition, it is easy to verify that E (U i ) = () (p s ) + (0) ( p s ) = p s E (W i ) = () (p t ) + (0) ( p t ) = p t : Since the trials are independent, it follows that, for any two distinct trials i and j, U i and W j must have zero covariance, that is, Cov (U i ; W j ) = 0 for i 6= j: For each trail i, because U i and W i cannot both appear to be one in the same trial, the product U i W j must always equal to zero so that E (U i W j ) = 0 Thus, Therefore, Cov (U i ; W i ) = E (U i W i ) E (U i ) E (W i ) = 0 p s p t = p s p t : 4

5 ! Cov (Y s ; Y t ) = Cov U i ; W j = Cov (U i ; W j ) = j= j= Cov (U i ; W i ) + X X Cov (U i ; W j ) = 0 + i6=j ( p s p t ) = np s p t : Example 4 (Refer to Example 3) Since p = 9 ; p = 6 ; p 3 = 8 ; it follows from Theorem that E (Y ) = np = (6) = E (Y ) = np = (6) = 6 E (Y 3 ) = np 3 = (6) = V ar (Y ) = np q = (6) = V ar (Y ) = np q = (6) = V ar (Y 3 ) = np 3 q 3 = (6) = Cov (Y ; Y ) = np p = (6) = Cov (Y ; Y 3 ) = np p 3 = (6) = Cov (Y ; Y 3 ) = np p 3 = (6) = : 5

JOINT PROBABILITY DISTRIBUTIONS

MTH/STA 56 JOINT PROBABILITY DISTRIBUTIONS The study of random variables in the previous chapters was restricted to the idea that a random variable associates a single real number with each possible outcome