Determine the cumulative distribution function of a binomial random variable with n=3 and p=1 /

Determine the cumulative distribution function of a binomial...

Key Concepts

Cumulative Distribution Function (CDF)

The CDF of a random variable gives the probability that the variable will take a value less than or equal to a specific number. In the context of discrete distributions like the binomial, it is calculated by summing the probabilities (given by the PMF) for all outcomes up to that number.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent Bernoulli trials, each with the same probability of success. It is characterized by two parameters: n, which is the number of trials, and p, which is the probability of success in each trial.

Probability Mass Function (PMF)

The PMF defines the probability that a discrete random variable is exactly equal to some value. For a binomial random variable, the PMF is given by the expression P(X = k) = (n choose k) * p^k * (1-p)^(n-k), which assigns a probability to each possible number of successes.

Transcript

00:04 Okay, we want to come up with the cumulative distribution function for the binomial random variable x, where n, the number of trials is three, and p to probability of success is one -fourth, which equals 0 .25.

00:24 First, i'd like to say, or i'd like us to find out, what is the probability? if we have three trials, what's the probability of zero successes? what's the probability of exactly one success? what's the probability of exactly two successes? and what is the probability of exactly three successes? we need these values before we try to create the cumulative distribution function.

01:00 The cumulative distribution function, we will call it f of x.

01:05 We'll talk more about that in a minute.

01:07 All right.

01:18 So with a little help from the probability applet, we're drawing the probability mass function.

01:28 This is not the cumulative distribution function.

01:30 This is the probability mass function of the binomial distribution parameters.

01:35 And is 3 .p is 0 .25.

01:38 Now, i need to know what is the probability of 0, 1, 2, and 3 .6 .5.

01:46 So the probability of x equaling 0 is 0 .4219, probability of exactly one success, also 0 .4219, probability of having exactly two successes, 0 .146, and a probability of exactly three successes, 0 .0156.

02:50 If you add up all these individual probabilities, you will get a total of one as you should.

02:57 Cumulative distribution function, f of x.

03:02 Okay.

03:04 F of zero.

03:16 F of zero means the probability of having zero or fewer successes at most zero successes.

03:26 So same thing as probability that x equals 0 .4219.

03:34 F of 1 cumulative distribution function.

03:41 So f of 1, the probability that we have one or less successes at most 1...