Mean of Binomial Distribution

The mean of the binomial distribution is also known as its expected value. It represents the average number of successes you would expect over many repetitions of a binomial experiment. To find the mean of a binomial distribution, you multiply the number of trials (n) by the probability of success in a single trial (p). Therefore, the mean of the binomial distribution formula is μ = np. In this article, you’ll learn how to derive the mean of a binomial distribution formula and how to apply the formula in solving problems.

Deriving the Mean of the Binomial Distribution

In statistics, students are often asked to prove that the mean of a binomial distribution is np. Here’s how to go about it:

Step 1: Define the expected value of a random variable X

By definition, the expected value formula is:

𝐸[𝑋] = $\sum _{x=0}^{n}x\cdot P(X=x)$

Step 2: Substitute the binomial probability mass function (PMF) in the expectation formula

The pmf of a binomial distribution is:

$P(X=x)=\left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x}$

Substituting in the expectation formula, we have:

$E\left[X\right]=\sum _{x=0}^{n}x\cdot \left(\genfrac{}{}{0ex}{}{n}{x}\right)p^x(1-p{)}^{n-x}$

Step 3: Handle the x=0 term

In the above expectation formula, we can see that the summation starts from 0. Let’s substitute 0 in the formula.

Thus, when x = 0, the term becomes $0\cdot \left(\genfrac{}{}{0ex}{}{n}{0}\right)p^0(1-p{)}^n=0$

This means that we should start the summation from 1 instead of 0.

Therefore, the expectation formula becomes:

$E\left[X\right]=\sum _{x=1}^{n}x\cdot \frac{n!}{x!(n-x)!}{p}^x(1-p{)}^{n-x}$

Recall:$\left(\genfrac{}{}{0ex}{}{n}{x}\right)=\frac{n!}{x!(n-x)!}$

Step 4: Simplify the Binomial coefficient

$x\cdot \left(\genfrac{}{}{0ex}{}{n}{x}\right)=x\cdot \frac{n!}{x!(n-x)!}=\frac{n!}{(x-1)!(n-x)!}=n\cdot \frac{(n-1)!}{(x-1)!\left(\right(n-1)-(x-1\left)\right)!}=n\left(\genfrac{}{}{0ex}{}{n-1}{x-1}\right)$

Step 5: Substitute the simplified form of the binomial coefficient in the expectation formula and factor out np

$E\left[X\right]=\sum _{x=1}^{n}n\left(\genfrac{}{}{0ex}{}{n-1}{x-1}\right)p^x(1-p{)}^{n-x}$

Factoring out np in the equation gives:

$E\left[X\right]=np\sum _{x1}^n\left(\genfrac{}{}{0ex}{}{n-1}{x-1}\right)p^{x-1}(1-p{)}^{n-x}$

Step 6: Apply the binomial theorem

Let k = x-1. Therefore, when x = 1, k = 0 and when x = n, then k = n-1

Thus, the summation becomes:

$\sum _{k=0}^{n-1}\left(\genfrac{}{}{0ex}{}{n-1}{k}\right)p^k(1-p{)}^{(n-1)-k}$

Now, using the binomial theorem, the above expansion corresponds to:

$(p+(1-p){)}^{n-1}$

= $1^{n-1}=1$

Therefore, $E\left[X\right]=np\cdot 1=np$. Hence the proof.

As such, the expected value of the binomial distribution is np.

How to Find the Mean of a Binomial Distribution

To find the mean of any binomial distribution, you need to apply the formula μ = np. Therefore, given any binomial problem, you need to follow these steps to calculate the expected value:

• Step 1: Identify the number of trials (n).
• Step 2: Identify the probability of success (p).
• Step 3: Multiply n by p to get the mean

However, if you’re working with a probability mass function (PMF) table, you need to follow these steps:

1. List all possible values of X, from 0 to n.
2. Write the probability of each value, P(X = x).
3. Multiply each value by its probability.
4. Add all these products using the formula: μ = Σx.P(X = x)

Solved Examples on Calculating the Mean of Binomial Distribution

Example 1

A class has a quiz with 15 independent multiple-choice questions. A student knows the correct answer 40% of the time. Calculate the mean of this binomial distribution.

Solution

From Example 1, we know that:

• The number of trials, n = 15
• The probability of success, p = 0.4

Applying the expected value formula, we have:

mean = np

= 15 × 0.4

= 6

Therefore, the student is expected to get 6 questions correct on average.

For this type of question, our free binomial probability calculator can also help find the expected value.

Example 2

A grocery store inspected crates of tomatoes and recorded the number of rotten tomatoes found in each crate. The results were:

• 95% of crates have 0 rotten tomatoes
• 2% of crates have 1 rotten tomato
• 2% of crates have 2 rotten tomatoes
• 1% of crates have 3 rotten tomatoes

Find the mean number of rotten tomatoes per crate.

Solution

Because this distribution does not follow a standard binomial form, we cannot use μ = n × p. Instead, we use the PMF method. The steps are as follows:

• Step 1: Create a Probability Mass Function Table

• Step 2: Multiply each x by its probability.

0 × 0.95 = 0
1 × 0.02 = 0.02
2 × 0.02 = 0.04
3 × 0.01 = 0.03

• Step 3: Add the products.

0 + 0.02 + 0.04 + 0.03 = 0.09

Thus, the mean number of rotten tomatoes per crate is 0.09.

Example 3

A random variable X represents the number of successes in 3 independent trials, where the probability of success in each trial is 0.4. The probability mass function (PMF) of X is given in the table below.

1. Show that this is a valid probability mass function.
2. Use the PMF table to find the mean (expected value) of $X$X.
3. Verify your answer in part (b) using the formula for the mean of a binomial distribution.

Solution

(i) Showing that this is a valid PMF

A probability mass function is valid if:

1. 0≤P(X=x)≤1 for all values of x
2. The sum of all probabilities is equal to 1

Check condition (1)

From the probability mass function table, all the values in p(X=x) are between 0 and 1. That is, 0.216,0.432,0.288,0.064 ∈ [0,1]

So condition (1) is satisfied.

Check condition (2)

The sum of all probabilities in the pmf shoud be equal to 1.

∑P(X=x)=0.216+0.432+0.288+0.064

=1.000

Since both conditions are satisfied, the table is a valid probability mass function.

(ii) Finding the Mean Using the PMF Table

The mean (expected value) of a discrete random variable is given by:

E(X)=∑x.P(X=x)

Therefore, substituting the values from the pmf in the formula, we have:

E(X)​=(0)(0.216)+(1)(0.432)+(2)(0.288)+(3)(0.064)

=0+0.432+0.576+0.192

=1.20​

Hence E(X) = 1.2

(iii) Verification Using the Binomial Mean Formula

Since X∼Binomial(n=3,p=0.4), the mean of the binomial distribution can be calculated as follows:

μ=np

=3×0.4

=1.2

This agrees with the value obtained in part (ii).