We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

AP Statistics : How to find standard deviation of a random variable

Study concepts, example questions & explanations for AP Statistics

Create An Account Create Tests & Flashcards

All AP Statistics Resources

4 Diagnostic Tests 140 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 :How To Find Standard Deviation Of A Random Variable

We have two independent, normally distributed random variables $\small X_1$ and $\small X_2$ such that $\small X_1$ has mean $\small 0$ and variance $\small 2$ and $\small X_2$ has mean $\small 0$ and variance $\small 1$ . What is the probability distribution of the difference of the random variables, $\small \small X_1-X_2$ ?

Possible Answers:

Normal distribution with mean $\small 1$ and variance $\small 2$ .

Normal distribution with mean $\small 1$ and variance $\small 3$ .

Normal distribution with mean $\small 0$ and variance $\small 3$ .

Normal distribution with mean $\small 0$ and variance $\small 1$ .

Correct answer:

Normal distribution with mean $\small 0$ and variance $\small 3$ .

Explanation:

任何一组随机变量的意思是additive in the sense that

$\small Mean(X_1+X_2)=Mean(X_1)+Mean(X_2)$

The difference is also additive, so we have

$\small Mean(X_1-X_2)=Mean(X_1)-Mean(X_2)$

This means the mean of $\small X_1-X_2$ is $\small 0$ .

The variance is additive when the random variables are independent, which they are in this case. But it's additive in the sense that for any real numbers $\small a,b$ (even when negative), we have

$\small Var(aX_1+bX_2)=a^2Var(X_1)+b^2Var(X_2)$ .

So for this difference, we have

$\small \small \small Var(X_1-X_2)=Var(X_1+(-1)X_2)=1^2Var(X_1)+(-1)^2Var(X_2)$

$\small =Var(X_1)+Var(X_2)=2+1=3$ .

So the mean and variance are $\small 0$ and $\small 3$ , respectively. In addition to that, $\small X_1-X_2$ is normally distributed because the sum or difference of any set of independent normal random variables is also normally distributed.

Report an Error

Example Question #1 :How To Find Standard Deviation Of A Random Variable

If $\small X_1$ and $\small X_2$ are two independent random variables with Var(X_1)=3 and Var(X_2)=3 , what is the standard deviation of the sum, $\small X_1+X_2?$

Possible Answers:

$\small \small \sqrt{18}$

${}\small \small \sqrt{3}$

$\small \sqrt{6}$

$\small 6$

Correct answer:

$\small \sqrt{6}$

Explanation:

If the random variables are independent, the variances are additive in the sense that

$\small Var(X+Y)=Var(X)+Var(Y)$ .

So then the variance of the sum is

$\small \small Var(X_1+X_2)=Var(X_1)+Var(X_2)=3+3=6$ .

The standard deviation is the square root of the variance, so we have

$\small SD(X_1+X_2)=\sqrt{6}$ .

Report an Error

Example Question #2 :How To Find Standard Deviation Of A Random Variable

Consider the discrete random variable $\small X$ that takes the following values with the corresponding probabilities:

$\small X=1$ with $\small P(X=1)=\frac{1}{8}$
$\small X=2$ with $\small P(X=2)=\frac{3}{8}$
$\small X=3$ with $\small P(X=3)=\frac{1}{2}$

Compute the probability $\small P(X\ge2)$ .

Possible Answers:

$\small \frac{1}{2}$

$\small \frac{1}{8}$

$\small \frac{7}{8}$

$\small \frac{1}{4}$

Correct answer:

$\small \frac{7}{8}$

Explanation:

This probability is simple to compute:

We want to add the probability that X is greater or equal to two. This means the probability that X=2 or X=3.

$\small \small P(X\ge 2)=P(X=2 \or X=3)$

Adding the necessary probabilities we arrive at the solution.

=P(X=2)+P(X=3)

$\small =\frac{3}{8}+\frac{1}{2}=\frac{7}{8}$

Report an Error

Example Question #1 :How To Find Standard Deviation Of A Random Variable

Consider the discrete random variable $\small X$ that takes the following values with the corresponding probabilities:

$\small X=1$ with $\small \small P(X=1)=\frac{1}{4}$
$\small X=2$ with $\small \small P(X=2)=\frac{1}{4}$
$\small X=3$ with $\small \small P(X=3)=\frac{1}{4}$
$\small X=4$ with $\small P(X=4)=\frac{1}{4}$

Compute the expected value of the distribution.

Possible Answers:

$\small 2$

$\small 2.5$

$\small 10$

$\small 3$

Correct answer:

$\small 2.5$

Explanation:

The expected value is computed as

$\small \mathbb{E}[X]=\small \sum_{x} xP(X=x)$

for any values of x that the random variable takes.

So we have

$\small \small \mathbb{E}[X]= \frac{1}{4}\cdot 1+\frac{1}{4}\cdot 2+\frac{1}{4}\cdot 3+\frac{1}{4}\cdot 4$

$\small =\frac{10}{4}=2.5$

Report an Error

Example Question #1 :How To Find Standard Deviation Of A Random Variable

The average number of calories in a Lick Yo' Lips lollipop is 210 , with a standard deviation of. The calories per lollipop are normally distributed, so what percent of lollipops have more than 225 calories?

Possible Answers:

$4.57\%$

$47.9\%$

$3.56\%$

$45.7\%$

$6.68\%$

Correct answer:

$6.68\%$

Explanation:

The random variable X = number of calories per lollipop, so the answer is

$P(\mbox{Number of Calories} > 225)$ or

$P(X > 225) = P(Z > \frac{(225-210)}{10}) = P(Z > 1.5) = 1- 0.9332 = 0.0668.$

Report an Error

Example Question #1 :How To Find Standard Deviation Of A Random Variable

Robert's work schedule for next week will be released today. Robert will work either 45, 40, 25, or 12 hours. The probabilities for each possibility are listed below:

45 hours: 0.3

40 hours: 0.2

25 hours: 0.4

12 hours: 0.1

What is the standard deviation of the possible outcomes?

Possible Answers:

11.07

122.61

32.7

13.5

35.8

Correct answer:

11.07

Explanation:

There are four steps to finding the standard deviation of random variables. First, calculate the mean of the random variables. Second, for each value in the group (45, 40, 25, and 12), subtract the mean from each and multiply the result by the probability of that outcome occurring. Third, add the four results together. Fourth, find the square root of the result.

$(45-32.7)^{2}(0.3)=45.38$

$(40-32.7)^{2}(0.2)=10.66$

$(25-32.7)^{2}(0.4)=23.72$

$(12-32.7)^{2}(0.1)=42.85$

45.38+10.66+23.72+42.85=122.61

$\sqrt{}122.61=11.07$

Report an Error

View AP Statistics Tutors

Haani
Certified Tutor

University of North Carolina at Chapel Hill, Bachelor in Arts, Economics. North Carolina State University at Raleigh, Master ...

View AP Statistics Tutors

Ping
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Computer Science. University of North Texas, Master of Arts, Education.

View AP Statistics Tutors

William
Certified Tutor

University of Toronto, Bachelor of Science, Criminology. University of Regina, Master of Arts, Applied Psychology.

All AP Statistics Resources

4 Diagnostic Tests 140 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in Seattle,SSAT Tutors in Dallas Fort Worth,ACT Tutors in Atlanta,Computer Science Tutors in Atlanta,Calculus Tutors in Denver,Computer Science Tutors in San Francisco-Bay Area,MCAT Tutors in Boston,Calculus Tutors in Chicago,SAT Tutors in Atlanta,称MCAT导师在费城

Popular Courses & Classes

GMAT考试Courses & Classes in Dallas Fort Worth,MCAT Courses & Classes in Boston,Spanish Courses & Classes in New York City,ACT Courses & Classes in Miami,Spanish Courses & Classes in Seattle,GMAT考试Courses & Classes in Chicago,GRE Courses & Classes in Philadelphia,SAT Courses & Classes in Los Angeles,GMAT考试Courses & Classes in San Francisco-Bay Area,SAT Courses & Classes in New York City

Popular Test Prep

GMAT考试Test Prep in Chicago,SAT Test Prep in Houston,SAT Test Prep in Washington DC,GMAT考试Test Prep in San Francisco-Bay Area,ACT Test Prep in Miami,LSAT Test Prep in Dallas Fort Worth,ISEE Test Prep in San Diego,SSAT Test Prep in Washington DC,GRE Test Prep in New York City,LSAT Test Prep in Chicago

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

*Full legal name

Company name

*电话号码

*Email address

Address

*Address1

Address2

*City

*State

*Zipcode

*Country

Complaint Details

*URL of copyrighted work (where your original material is located)

*请描述版权工作的马y be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

*Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

AP Statistics : How to find standard deviation of a random variable

Study concepts, example questions & explanations for AP Statistics

All AP Statistics Resources

Example Questions

Example Question #1 :How To Find Standard Deviation Of A Random Variable

Example Question #1 :How To Find Standard Deviation Of A Random Variable

Example Question #2 :How To Find Standard Deviation Of A Random Variable

Example Question #1 :How To Find Standard Deviation Of A Random Variable

Example Question #1 :How To Find Standard Deviation Of A Random Variable

Example Question #1 :How To Find Standard Deviation Of A Random Variable

All AP Statistics Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Find the Best Tutors

Email address:
Your name:
Feedback: