Question

What is the formula for the expected number of successes in a binomial experiment with n trials and probability of success​ p?

A. Upper E (Upper X )equalsE(X)=StartRoot np (1 minus p )EndRootnp(1−p)
B. Upper E (Upper X )equalsE(X)=p Superscript npn
C. Upper E (Upper X )equals (1 minus p )Superscript nE(X)=(1−p)n(1 minus p )Superscript n(1−p)n
D. Upper E (Upper X )equalsE(X)=np

Answer 1

`X \sim Binom(n, p)`

The probability mass function for the Binomial distribution is given as:

`P(X)=(nCx)(p)^x (1-p)^(n-x)`

Where (nCx) means combinatory and it's given by this formula:

`nCx=(n!)/((n-x)! x!)`

The expected value is given by this formula:

`E(X) = np`

For this reason the best option is:

D. E (X )=np

Explanation:

Previous concepts

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

Solution to the problem

Let X the random variable of interest, on this case we now that:

`X \sim Binom(n, p)`

The probability mass function for the Binomial distribution is given as:

`P(X)=(nCx)(p)^x (1-p)^(n-x)`

Where (nCx) means combinatory and it's given by this formula:

`nCx=(n!)/((n-x)! x!)`

The expected value is given by this formula:

`E(X) = np`

For this reason the best option is:

D. E (X )=np