Binomial Distribution in Statistics

Expectation of X in Binomial Distribution

Let X be a random variable, it is said to follow binomial distribution if it follows the following probability mass function. And it can have only non-negative values. The binomial distribution is a discrete probability distribution.

PDF-of-Binomial-Distribution

Binomial distribution is used to model problems, for instance,  getting  number of success after certain  number of random experiments or trials.

Example-

Problem- If there are four coins which are tossed simultaneously, calculate the probability of getting at least  two heads.

Solution-

Probability of getting head of a coin = 1/2

Probability of getting tail of a coin= 1/2

i.e

p=1/2 and q=1/2

probability of getting x heads in trials of 4 coins is

https://www.postnetwork.co/wp-content/uploads/Binomial-Distribution-Solution.png

Therefore probability of getting at least two heads are

Binomial-DistributionSolution-2

Mean or Expectation  of Binomial Distribution-

Expectation or mean of binomial distributed random variable X is

Expectation of X in Binomial Distribution

Variance of binomial distributed random variable X is

Binomial-Distribution-Variance

Python Code for Binomial Distribution


from scipy.stats import binom
import numpy as np
import matplotlib.pyplot as plt
# Let lambda=np=5
x = np.arange(0,10)
n=50
p=0.10
plt.plot(x, binom.pmf(x, n, p))
plt.savefig(“binom.png”)


Output would be
Binomial Distribution For Small n

Conclusion

In this post, I have explained about binomial  distribution. Hope you will understand and apply it.

References

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