MULTINOMIAL DISTRIBUTION ASSIGNMENT HELP

The multinomial distribution is a probability density function used to calculate probabilities of multinomial experiments where the number of outcomes is more than two.

Conditions

1. The experiment has independent trials
2. Each trial can result in collectively exhaustive and mutually exclusive outcomes E1, E2, …, Ek.
3. The probability of each outcome Ei on every trial is π_i, i = 1, 2, …, k

Let,
X1 = No. of trials we get E1
X2 = No. of trials we get E2
…
Xk = No. of trials we get Ek

Hence the random variable X = (X1, X2, …,Xk) has a multinomial distribution with parameterπ and index n and denote it as X ~ Mult(n,π).

P={n!/(n₁!*n₂!*…*n_k!)} * (p₁^n₁)*(p₂^n₂)*…*p_k^n_k

Example

Let us assume you have an urn with 10 coloured balls in it.3 of the balls are green,2 are red and the rest are blue. Randomly pick 4 balls from the urn with replacement. Find the probability of picking only 2 blue and 2 green balls.

Solution

From the above statement it then follows that;
n = 4 trials, n1= 2 green balls
n2= 0 red balls
n3 = 2 blue balls
p1= 3/10 = 0.3
p2 = 2/10 = 0.2
p3= (10-(3+2))/10 = 5/10 = 0.5

Next, we substitute these values into the formula for multinomial distribution;
P = 4!/(2!*0!*2!) (0.3)²*(0.2)⁰*(0.5)² = 0.135
The probability of picking 2 blue and 2 green balls from an urn with replacement is 0.135.