### Chapter 4. Probability Distributions: Discrete Probability Distributions

### The Binomial Probability Distribution

Binomial Experiment

In a **binomial **experiment:

- We conduct #n# independent Bernoulli trials.
- The probability of success #p# is the same for each trial.
- The variable of interest #X# is the total number of successes observed.

Binomial Distribution

Let #X# be the number of successes among #n# trials in a binomial experiment, then #X# is a **binomial random variable **with range: \[R(X)=\{0,1,\ldots,n\}\]

We say that #X# is **binomially distributed **with parameters #n# and #p# and write this as: \[X \sim B(n,p)\]

Suppose we flip a coin #12# times and define a success as the coin coming up Heads.

Let #X# be the number of successes among the #12# inquiries.

Then #X# is *binomially *distributed with #n=12# and #p=\mathbb{P}(\textit{Heads}) = 0.5#: \[X\sim B(12,0.5)\]

Suppose that #21\%# of people in a large population are smokers. Choose #100# people at random and ask them: "Are you a smoker"? Define a person answering "Yes" as a success.

Let #Y# be the number of successes among the #100# observations.

Then #Y# is *binomially *distributed with #n=100# and #p=0.21#: \[Y\sim B(100, 0.21)\]

#\phantom{0}#

Computation of Binomial Probabilities with Statistical Software

Let #X# be a *binomial *random variable with parameters #n# and #p#.

To compute #\mathbb{P}(X=x)# in Excel, make use of the following function:

BINOM.DIST(x, n, p, cumulative)

x: The number of successes.n: The number of trials.p: The probability of success for each trial.cumulative: A logical value that determines the form of the function.

- TRUE - uses the cumulative distribution (at most
xsuccesses), #\mathbb{P}(X \leq x)#- FALSE - uses the probability mass function (exactly
xsuccesses), #\mathbb{P}(X = x)#

To compute #\mathbb{P}(X=x)# in R, make use of the following function:

dbinom(x, size, prob)

x: The number of successes.size: The number of trials.prob: The probability of success for each trial.

*binomial*distribution with #n=12# and #p=0.8#.

Compute #\mathbb{P}(X = 11)#. Round your answer to #3# decimal places.

There are a number of different ways we can calculate #\mathbb{P}(X = 11)#. Click on one of the panels to toggle a specific solution.

To calculate #\mathbb{P}(X = 11)# in Excel, make use of the following function:

Thus, to calculate #\mathbb{P}(X = 11)#, run the following command:

BINOM.DIST(x, n, p, cumulative)

x: The number of successes.n: The number of trials.p: The probability of success of each trial.cumulative: A logical value that determines the form of the function.

- TRUE - uses the cumulative distribution (at most
xsuccesses), #\mathbb{P}(X \leq x)#- FALSE - uses the probability mass function (exactly
xsuccesses), #\mathbb{P}(X = x)#

\[= \text{BINOM.DIST}(11, 12, 0.8, \text{FALSE})\]

This gives:

\[\mathbb{P}(X = 11) = 0.206\]

To calculate #\mathbb{P}(X = 11)# in R, make use of the following function:

Thus, to calculate #\mathbb{P}(X = 11)#, run the following command:

dbinom(x, size, prob)

x: The number of successes.size: The number of trials.prob: The probability of success of each trial.

\[\text{dbinom}(x = 11, size = 12, prob = 0.8)\]

This gives:

\[\mathbb{P}(X = 11) = 0.206\]

#\phantom{0}#

Computation of Cumulative Binomial Probabilities with Statistical Software

Let #X# be a *binomial *random variable with parameters #n# and #p#.

To calculate *cumulative *probabilities for a *binomial* distribution in Excel, make use of the following function:

BINOM.DIST(x, n, p, cumulative)

x: The number of successes.n: The number of trials.p: The probability of success for each trial.cumulative: A logical value that determines the form of the function.

- TRUE - uses the cumulative distribution (at most
xsuccesses), #\mathbb{P}(X \leq x)#- FALSE - uses the probability mass function (exactly
xsuccesses), #\mathbb{P}(X = x)#

To calculate *cumulative *probabilities for a *binomial* distribution in R, make use of the following function:

pbinom(q, size, prob)

q: The number of successes.size: The number of trials.prob: The probability of success for each trial.

There are a number of different ways we can calculate #\mathbb{P}(X \leq 7)#. Click on one of the panels to toggle a specific solution.

To calculate #\mathbb{P}(X \leq 7)# in Excel, make use of the following function:

BINOM.DIST(x, n, p, cumulative)

x: The number of successes.n: The number of trials.p: The probability of success of each trial.cumulative: A logical value that determines the form of the function.

- TRUE - uses the cumulative distribution (at most
xsuccesses), #\mathbb{P}(X \leq x)#- FALSE - uses the probability mass function (exactly
xsuccesses), #\mathbb{P}(X = x)#

Thus, to calculate #\mathbb{P}(X \leq 7)#, run the following command:

\[= \text{BINOM.DIST}(7, 10, 0.8, \text{TRUE})\]

This gives:

\[\mathbb{P}(X \leq 7) = 0.322\]

To calculate #\mathbb{P}(X \leq 7)# in R, make use of the following function:

Thus, to calculate #\mathbb{P}(X \leq 7)#, run the following command:

pbinom(q, size, prob)

q: The number of successes.size: The number of trials.prob: The probability of success of each trial.

\[\text{pbinom}(q = 7, size = 10, prob = 0.8)\]

This gives:

\[\mathbb{P}(X \leq 7) = 0.322\]

#\text{}#

Mean, Variance, and Standard Deviation of a Binomial Random Variable

Let #X# be a *binomially* distributed random variable with parameters #n# and #p#.

Then the **expected value** of #X# calculated with the following formula: \[\mu = n\cdot p\]

The **variance **of #X# is calculated with the following formula:\[\sigma^2 = n\cdot p \cdot (1-p)\]

And the **standard deviation **of #X# is calculated with the following formula:\[\sigma = \sqrt{n\cdot p \cdot (1-p)}\]

The

*expected value*of a binomial random variable #X\sim B(n,p)# is calculated as follows:

\[\begin{array}{rcl}

\mu&=&n \cdot p \\\\

&=& 21 \cdot 0.30 \\\\

&=& 6.30

\end{array}\]

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