Last modified: June 23, 2024
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Binomial Distribution (Discrete)
A discrete random variable X follows a binomial distribution if it represents the number of successes in a fixed number of Bernoulli trials with the same probability of success. The binomial distribution is denoted as X∼Binomial(n,p), where n is the number of trials and p is the probability of success.
Probability Mass Function (PMF)
The PMF of a binomial distribution is given by:
P(X=k)=(nk)pk(1−p)n−k
where k∈0,1,2,…,n and (nk)=n!k!(n−k)! is the binomial coefficient.
Cumulative Distribution Function (CDF)
The CDF of a binomial distribution is given by:
F(k)=P(X≤k)=k∑i=0(ni)pi(1−p)n−i
Expected Value and Variance
The expected value (mean) of a binomial distribution is the product of the number of trials and the probability of success:
E[X]=np
The variance of a binomial distribution is the product of the number of trials, the probability of success, and the probability of failure:
Var(X)=np(1−p)
Moment Generating Functions and Moments
The moment generating function (MGF) of a binomial distribution is:
MX(t)=E[etX]=(pet+1−p)n
To find the n-th moment, we take the n-th derivative of the MGF with respect to t and then evaluate it at t=0:
E[Xn]=dnMX(t)dtn|t=0
- First Moment (Mean):
E[X]=np
- Second Moment (Variance + Mean^2):
E[X2]=np(1−p)+n2p2
Example: Company Email Security
A company's IT department reports that 90% of their employees follow proper email security protocols. If 10 employees are randomly chosen for an audit.
Given:
- Probability of success (following protocol)
p = 0.9 - Number of trials (employees chosen)
n = 10
I. What is the probability exactly 7 of them follow the security protocols?
For exactly 7 employees:
P(X=7)=(107)(0.9)7(0.1)10−7=0.0574
II. What is the probability at least 8 of them follow the security protocols?
For at least 8 employees:
P(X≥8)=10∑k=8(10k)(0.9)k(0.1)10−k=0.9298
III. What is the probability fewer than 5 employees follow the protocols?
For fewer than 5 employees:
P(X<5)=4∑k=0(10k)(0.9)k(0.1)10−k=0.0001
Applications
Binomial distributions are used in a variety of applications, such as quality control, risk analysis, and survey sampling, where the probability of success and the number of trials are known or can be estimated.