MA 320-001: Introductory Probability - Mathematicsdmu228/ma320/lectures/sec3.2part3.pdfBernoulli...

MA 320-001: Introductory Probability

David Murrugarra

Department of Mathematics,University of Kentucky

http://www.math.uky.edu/~dmu228/ma320/

Spring 2017

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 1 / 12

Bernoulli Experiment

A Bernoulli experiment is a random experiment, the outcome ofwhich can be classified in one of two mutually exclusive andexhaustive ways–say, success or failure.

Let X be a random variable associated with a Bernoulli trial by definingit as follows:

X (success) = 1 and X (failure) = 0.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 2 / 12

Bernoulli Experiment

A Bernoulli experiment is a random experiment, the outcome ofwhich can be classified in one of two mutually exclusive andexhaustive ways–say, success or failure.

Let X be a random variable associated with a Bernoulli trial by definingit as follows:

X (success) = 1 and X (failure) = 0.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 2 / 12

Bernoulli Trials

A sequence of Bernoulli trials occur when a Bernoulli experiment isperformed several independent times so that the probability of successremains the same from trial to trial.

Let p denote the probability of success in each trial and q = 1− p theprobability of failure.

ExampleConsider the experiment of flipping a fair coin five independent times.

The probability of heads on any one toss is 1/2.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 3 / 12

Bernoulli Trials

A sequence of Bernoulli trials occur when a Bernoulli experiment isperformed several independent times so that the probability of successremains the same from trial to trial.

Let p denote the probability of success in each trial and q = 1− p theprobability of failure.

ExampleConsider the experiment of flipping a fair coin five independent times.

The probability of heads on any one toss is 1/2.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 3 / 12

Bernoulli Trials

A sequence of Bernoulli trials occur when a Bernoulli experiment isperformed several independent times so that the probability of successremains the same from trial to trial.

Let p denote the probability of success in each trial and q = 1− p theprobability of failure.

ExampleConsider the experiment of flipping a fair coin five independent times.

The probability of heads on any one toss is 1/2.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 3 / 12

Binomial Probabilities

We shall be interested in the probability that in n Bernoulli trials thereare exactly j successes.

DefinitionGiven an n Bernoulli trial with probability p of success, the probabilityof exactly j successes is denoted by b(n,p, j).

ExampleCalculate b(3,p,2).

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 4 / 12

Binomial Probabilities

We shall be interested in the probability that in n Bernoulli trials thereare exactly j successes.

DefinitionGiven an n Bernoulli trial with probability p of success, the probabilityof exactly j successes is denoted by b(n,p, j).

ExampleCalculate b(3,p,2).

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 4 / 12

Binomial Probabilities

We shall be interested in the probability that in n Bernoulli trials thereare exactly j successes.

DefinitionGiven an n Bernoulli trial with probability p of success, the probabilityof exactly j successes is denoted by b(n,p, j).

ExampleCalculate b(3,p,2).

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 4 / 12

Binomial Probabilities

Let the random variable X equal the number of observed successes inn Bernoulli trials, then the possible values of X are 0,1,2, . . . ,n.

If x successes occur, where x = 0,1,2, . . . ,n, then n − x failuresoccur. The number of ways of selecting x positions for the x successesin the n trial is (

nx

)=

n!x!(n − x)!

.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 5 / 12

Binomial Probabilities

Let the random variable X equal the number of observed successes inn Bernoulli trials, then the possible values of X are 0,1,2, . . . ,n.

If x successes occur, where x = 0,1,2, . . . ,n, then n − x failuresoccur.

The number of ways of selecting x positions for the x successesin the n trial is (

nx

)=

n!x!(n − x)!

.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 5 / 12

Binomial Probabilities

Let the random variable X equal the number of observed successes inn Bernoulli trials, then the possible values of X are 0,1,2, . . . ,n.

If x successes occur, where x = 0,1,2, . . . ,n, then n − x failuresoccur. The number of ways of selecting x positions for the x successesin the n trial is (

nx

)=

n!x!(n − x)!

.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 5 / 12

Binomial Distribution

f (x) =(

nx

)px(1− p)n−x , x = 0,1,2, . . . ,n.

These probabilities are called binomial probabilities, and the randomvariable X is said to have a binomial distribution.

A binomial distribution will be denoted by the symbol b(n,p), and wethat the distribution of X is b(n,p).The constants n and p are called theparameters of the binomial distribution.

∑x∈S

f (x) = 1

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 6 / 12

Binomial Distribution

f (x) =(

nx

)px(1− p)n−x , x = 0,1,2, . . . ,n.

These probabilities are called binomial probabilities, and the randomvariable X is said to have a binomial distribution.

A binomial distribution will be denoted by the symbol b(n,p), and wethat the distribution of X is b(n,p).The constants n and p are called theparameters of the binomial distribution.

∑x∈S

f (x) = 1

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 6 / 12

Binomial Distribution

f (x) =(

nx

)px(1− p)n−x , x = 0,1,2, . . . ,n.

These probabilities are called binomial probabilities, and the randomvariable X is said to have a binomial distribution.

A binomial distribution will be denoted by the symbol b(n,p), and wethat the distribution of X is b(n,p).The constants n and p are called theparameters of the binomial distribution.

∑x∈S

f (x) = 1

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 6 / 12

Binomial Distribution

Figure: Binomial density function.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 7 / 12

Binomial Distribution

f (x) =(

nx

)px(1− p)n−x , x = 0,1,2, . . . ,n.

µ = E(X ) = np.

σ2 = npq

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 8 / 12

Binomial Distribution

f (x) =(

nx

)px(1− p)n−x , x = 0,1,2, . . . ,n.

µ = E(X ) = np.

σ2 = npq

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 8 / 12

Cumulative Distribution Function

The cumulative distribution function or, more simply, thedistribution function of the random variable X is

F (x) = P(X ≤ x), −∞ < x <∞,

For the binomial distribution the distribution function is defined by

F (x) = P(X ≤ x) =bxc∑y=0

(ny

)py (1− p)n−y

where bxc is the floor or greatest integer less than or equal to x .

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 9 / 12

Binomial Distribution

Figure: Binomial distribution cdf.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 10 / 12

Binomial Distribution

ExampleConsider the experiment of flipping a fair coin six independent times.What is the probability that exactly three heads turn up?

b(6,0.5,3) =(

63

)(12

)3(12

)3

= 0.3125.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 11 / 12

Binomial Distribution

ExampleConsider the experiment of flipping a fair coin six independent times.What is the probability that exactly three heads turn up?

b(6,0.5,3) =(

63

)(12

)3(12

)3

= 0.3125.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 11 / 12

Binomial Distribution

ExampleConsider the experiment of rolling a fair die four independent times.What is the probability that exactly one six?

We this experiment as a 4 Bernoulli trials withsuccess: "rolling a 6" andfailure: "rolling some number other than 6".

Then p = 1/6.

b(4,1/6,1) =(

41

)(16

)1(56

)3

= 0.386.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 12 / 12

Binomial Distribution

ExampleConsider the experiment of rolling a fair die four independent times.What is the probability that exactly one six?

We this experiment as a 4 Bernoulli trials withsuccess: "rolling a 6" andfailure: "rolling some number other than 6".

Then p = 1/6.

b(4,1/6,1) =(

41

)(16

)1(56

)3

= 0.386.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 12 / 12

Binomial Distribution

ExampleConsider the experiment of rolling a fair die four independent times.What is the probability that exactly one six?

We this experiment as a 4 Bernoulli trials withsuccess: "rolling a 6" andfailure: "rolling some number other than 6".

Then p = 1/6.

b(4,1/6,1) =(

41

)(16

)1(56

)3

= 0.386.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 12 / 12

Binomial Distribution

ExampleConsider the experiment of rolling a fair die four independent times.What is the probability that exactly one six?

We this experiment as a 4 Bernoulli trials withsuccess: "rolling a 6" andfailure: "rolling some number other than 6".

Then p = 1/6.

b(4,1/6,1) =(

41

)(16

)1(56

)3

= 0.386.

David Murrugarra (University of Kentucky) MA 320: Section 3.2 Spring 2017 12 / 12