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Statistics / Statistics for Engineers and Scientists 4 / Chapter 4.1 / Problem 1E

Statistics for Engineers and Scientists | 4th Edition | ISBN: 9780073401331 | Authors: William Navidi

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After scoring a touchdown, a football team may elect to

Chapter 4, Problem 1E

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QUESTION:

After scoring a touchdown, a football team may elect to attempt a two-point conversion, by running or passing the ball into the end zone. If successful, the team scores two points. For a certain football team, the probability that this play is successful is 0.40.

a. Let X = 1 if successful, X = 0 if not. Find the mean and variance of \(X\).

b. If the conversion is successful, the team scores 2 points; if not the team scores 0 points. Let \(Y\) be the number of points scored. Does \(Y\) have a Bernoulli distribution? If so, find the success probability. If not, explain why not.

c. Find the mean and variance of \(Y\) .

Equation Transcription:

$X$

$Y$

Text Transcription:

X

Y

Questions & Answers

QUESTION:

After scoring a touchdown, a football team may elect to attempt a two-point conversion, by running or passing the ball into the end zone. If successful, the team scores two points. For a certain football team, the probability that this play is successful is 0.40.

a. Let X = 1 if successful, X = 0 if not. Find the mean and variance of \(X\).

b. If the conversion is successful, the team scores 2 points; if not the team scores 0 points. Let \(Y\) be the number of points scored. Does \(Y\) have a Bernoulli distribution? If so, find the success probability. If not, explain why not.

c. Find the mean and variance of \(Y\) .

Equation Transcription:

$X$

$Y$

Text Transcription:

X

Y

ANSWER:

Answer:

Step 1 of 3:

(a)

In this question, we are asked to find the mean and variance of $X$ .

Where $X$ is random variable of team scoring successfully two points in the football game.

For a certain football team, the probability that this play is successful is $0.40$ .

Let $X\ =\ 1$ if successful, $X\ =\ 0$ if not.

If the random event results in success, then $X\ =\ 1$ . Otherwise $X\ =\ 0$ . It follows that $X$ is a discrete random variable, with probability mass function (PMF) defined by

$p(x)\ =\ p(0)\ =\ P(X\ =\ 0)\ =\ 1-p$ $x\ =\ 0$

$p(x)\ =\ p(1)\ =\ P(X\ =\ 1)\ =\ p$ $x\ =\ 1$

$p(x)\ =\ 0$ for any value of x other than 0 or 1

The random variable $X$ is said to have the Bernoulli distribution with parameter $p$ .

And the notation is $X$ ∼ Bernoulli( $p$ ).

Mean $({\mu{}}_{X}$ ) of a Bernoulli random variable

${\mu{}}_{X}\ =\ p$ ………….(1)

Where $p$ is probability of success of any event which is given in the question.

$p\ =\ 0.40$

Substitute the value of $p$ into the equation (1)

${\mu{}}_{X}\ =\ 0.40$

Variance $({\sigma{}}_{X}^{2})$ of a Bernoulli random variable

${\sigma{}}_{X}^{2}\ =\ p(1\ -\ p)$ ………….(2)

$p\ =\ 0.40$

Substitute the value of $p$ into the equation (2)

${\sigma{}}_{X}^{2}\ =\ 0.40(1\ -\ 0.40)\ =\ 0.24$

Hence mean and variance of $X$ is $0.40$ and $0.24$ respectively.

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