Mean and Variance of Binomial Distribution

Mean or Expected value of a binomial distribution is given by the following formula:

Mean = μ = np

and variance or measure of a binomial distribution is given by the following formula:

Variance = σ² = np(1-p)

where, 

• n is Total Number of Trials
• p is Probability of Success

Important Things to Remember about Binomial Distribution

There are some important things related to binomial distribution to which we need to pay more attention. Some of those things are as follows:

• Binomial distribution is a legitimate probability distribution since

[Tex]\bold{\sum_{r=0}^n P(X=r)=\sum_{r=0}^n nCrq^{n-r}p^r=(q+p)^n=1}[/Tex]

• Mean of the Binomial Distribution is given by: 

[Tex]\bold{E(x)=\sum_{r} x_{r}p_{r}=np}[/Tex]

and

[Tex]\bold{E(x^2)=\sum_{r}x_{r}^2p_{r}}[/Tex]

• Variance of the Binomial Distribution is given by: 

[Tex]\bold{Var(x)=E(x^2)-{{E(x)}}^2=npq}[/Tex]

Generalization of Bernoulli’s Distribution: Multinomial Distribution

If A₁, A₂, . . , A_k are exhaustive and mutually exclusive events associated with a random experiment such that, P(A_i occurs) = p_i where, 

p₁ + p₂ +. . . + p_k = 1, and if the experiment is repeated n times, then the probability A₁ occurs r₁ times, A₂ occurs r₂ times, . . . . , A_k occurs r_k times is given by:

P_n(r₁, r_{2, . . . ,} r_k) = [Tex]\frac{n!}{r_{1}! r_{2}! . . . r_{k}!} \ p_{1}^{r_{1}}\times p_{2}^{r_{2}}\times. . .\times p_{k}^{r_{k}}[/Tex]

where,

• r₁​ + r₂ ​+ …+ r_k ​= n    

Proof:

r₁ trials in which the event A₁ occurs can be chosen from the n trials ⁿC_r ways. The remaining (n – r₁) trials are left over for the other events. 

r₂ trials in which the event A₂ occurs can be chosen from the (n – r₁) trials in ^{(n – r1)}C_r2  ways.

r₃ trials in which the event A₃ occurs can be chosen from the (n – r₁ – r₂) trials in ^{(n – r1 – r2)}C_r3 ways, and so on.

Therefore, the number of ways in which the events A₁​, A₂​, …, A_k can happen:

ⁿ​C_r1​ ​× ^{(n − r1}​⁾C_r2 ​​× ^{(n −r1 ​− r2​)}C_r3​​ × ^{(n−r1​ − r1​ – …− rk ​− 1)}C_rk​​ =  n!/(r₁!r₂! . . . r₃!)

Consider any one of the above ways in which the events A₁, A₂, . . ., A_k occurs.

Since, n trials are independent, r₁ + r₂ + . . . +r_k trials are also independent.

∴ P(A₁ occurs r₁ times, A₂ occurs r₂ times, . . . , A_k occurs r_k times) = p₁ r₁ × p₂r₂ × . . . × p_k r_k 

Since the ways in which the events happen are mutually exclusive, the required probability is given by

P_n (r₁ , r₂ , . . . , r_k ) =[Tex] \frac{n!}{r_{1}! r_{2}! . . . r_{k}!}\times \ p_{1} ^{r_{1}}\times p_{2}^{r_{2}}\times… \times p_{k}^{r_{k}}[/Tex]

Read More,

• Binomial Theorem
• Binomial Random Variables
• Binomial Standard Deviation

Bernoulli Trials and Binomial Distribution are the fundamental topics in the study of probability and probability distributions. Bernoulli’s Trials are those trials in probability where only two possible outcomes are Success and Failure or True and False. Due to this fact of two possible outcomes, it is also called the Binomial Trial.

Binomial Distribution is the sequence of independent experiments with each experiment being a binomial trial. In this article, we are going to discuss the Bernoulli Trials in detail with the related theorems as well. Also, we will study the Binomial Distribution after the understanding Bernoulli Trial.