Binomial Distribution and Binomial Random Variables | Probability Mass Function, Probability Theory

What is the binomial distribution and what are binomial random variables? In today’s video probability theory lesson, we give a thorough introduction to the binomial distribution and binomial random variables. We’ll discuss its expected value and variance, and we will go over the probability mass function of the binomial distribution (PMF) and the cumulative distribution function (CDF)!

We offer “lite” proofs of the expected value and variance of the binomial distribution, I say they are lite proofs because they would be valid proofs, but we didn’t necessarily fill in every last detail. We’ll also go over Bernoulli random variables and the probability mass function of the Bernoulli distribution, because we can think of Binomial random variables as the sum of independent and identically distributed Bernoulli random variables, which is actually very helpful!

We actually begin with the expected value and variance of a Bernoulli random variable, as these discoveries help us find the expected value and variance of a binomial random variable. In fact, a Bernoulli random variable is a binomial random variable, just a special kind! Watch the full lesson to see why that is the case!

A binomial random variable gives the number of successes in n independent trials of some experiment, where the probability of success is p (and thus the probability of failure is (1-p). The probability mass function of such a random variable involves a binomial coefficient, which is where the name “binomial distribution” comes from. We’ll see why the binomial distribution has this particular PMF by going through an example.

SOLUTION TO PRACTICE PROBLEM:

BEFORE READING THE SOLUTION - you should try to find E(X) and Var(X) as well, good practice!

Drug X cures the disease smithefallitis in 18% of patients who take it, so if we define curing the disease to be success, then p = 0.18. If 50 people take Drug X, that’s 50 trials of administering the drug. So n = 50. Thus, since the conditions of patients taking the drug are independent, we can say that the random variable X, giving the number of cured patients in the sample of 50, is binomial with parameters n = 50 and p = 0.18.

We want P(X=12), the probability that exactly 12 patients are cured. Using our PMF, this is equal to (50 choose 12)*(0.18)^12*(0.82)^38 = [50!/(12!*38!)]*(0.18^12)*(0.82^38). This is approximately 0.0745, that’s a 7.45% chance. The binomial coefficient there, 50 choose 12, is equal to 121,399,651,100, in case you were curious about that part.

The probability that X is less than 3 is the probability that X is less than or equal to 2, which is P(X=0) + P(X=1) + P(X=2) = 0.00004906 + 0.00053842 + 0.00289565 = 0.003483, or about 0.3483%. Not very likely! This is because, since the probability of success is nearly one fifth, it’s very unlikely we’d have only 0, 1, or 2 successes in 50 trials. Since p is nearly one fifth, we’d expect closer to 10 successes in 50 trials. Sure enough, the expected value of X, E(X), is equal to 50*0.18 = 9, so as we thought, close to 10! The variance, Var(X) is 50*0.18*0.82 = 7.38.

By the law of complement, since P(X is less than 3) is about 0.3484%, the rest of the probability must be where X is greater than or equal to 3, so the probability that AT LEAST 3 patients are cured (as in, X is greater than or equal to 3) is 100% - 0.3484% = 99.6516%.

If you are preparing for Probability Theory or in the midst of learning Probability Theory, you might be interested in the textbook I used in my Probability Theory course, called "A First Course in Probability Theory" by Sheldon Ross. Check out the book and see if it suits your needs! You can purchase the textbook using the affiliate link below which costs you nothing extra and helps support Wrath of Math!

PURCHASE THE BOOK: https://amzn.to/31mXEjr



I hope you find this video helpful, and be sure to ask any questions down in the comments!

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The outro music is by a favorite musician of mine named Vallow, who, upon my request, kindly gave me permission to use his music in my outros. I usually put my own music in the outros, but I love Vallow's music, and wanted to share it with those of you watching. Please check out all of his wonderful work.

Vallow Bandcamp: https://vallow.bandcamp.com/
Vallow Spotify: https://open.spotify.com/artist/0fRtu...
Vallow SoundCloud:   / benwatts-3  
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