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This module explores the Law of Large Numbers, the phenomenon where an experiment performed many times will yield cumulative results closer and closer to the theoretical mean over time.

The expected value is often referred to as the "long-term"average or mean . This means that over the long term of doing an experiment over and over, you would expect this average.

The mean of a random variable X is μ . If we do an experiment many times (for instance, flip a fair coin, as Karl Pearson did, 24,000 times and let X = the number of heads) and record the value of X each time, the average is likely to get closer and closer to μ as we keep repeating the experiment. This is known as the Law of Large Numbers .

To find the expected value or long term average, μ , simply multiply each value of the random variable by its probability and add the products.

A step-by-step example

A men's soccer team plays soccer 0, 1, or 2 days a week. The probability that they play 0 days is 0.2, the probability that they play 1 day is 0.5, and the probability that they play 2 days is 0.3. Find the long-term average, μ , or expected value of the days per week the men's soccer team plays soccer.

To do the problem, first let the random variable X = the number of days the men's soccer team plays soccer per week. X takes on the values 0, 1, 2. Construct a PDF table, adding a column xP(x) . In this column, you will multiply each x value by its probability.

This table is called an expected value table. The table helps you calculate the expected value or long-term average.
Expected value table
x P(x) x P(x)
0 0.2 (0)(0.2) = 0
1 0.5 (1)(0.5) = 0.5
2 0.3 (2)(0.3) = 0.6

Add the last column to find the long term average or expected value: (0)(0.2)+(1)(0.5)+(2)(0.3)= 0 + 0.5 + 0.6 = 1.1 .

The expected value is 1.1. The men's soccer team would, on the average, expect to play soccer 1.1 days per week. The number 1.1 is the long term average or expected value if the men's soccer team plays soccer week after week after week. We say μ=1.1

Find the expected value for the example about the number of times a newborn baby's crying wakes its mother after midnight. The expected value is the expected number of times a newborn wakes its mother after midnight.

You expect a newborn to wake its mother after midnight 2.1 times, on the average.
x P(X) x P(X)
0 P(x=0) = 2 50 (0) ( 2 50 ) = 0
1 P(x=1) = 11 50 (1) ( 11 50 ) = 11 50
2 P(x=2) = 23 50 (2) ( 23 50 ) = 46 50
3 P(x=3) = 9 50 (3) ( 9 50 ) = 27 50
4 P(x=4) = 4 50 (4) ( 4 50 ) = 16 50
5 P(x=5) = 1 50 (5) ( 1 50 ) = 5 50

Add the last column to find the expected value. μ = Expected Value = 105 50 = 2.1

Go back and calculate the expected value for the number of days Nancy attends classes a week. Construct the third column to do so.

2.74 days a week.

Suppose you play a game of chance in which five numbers are chosen from 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. A computer randomly selects five numbers from 0 to 9 with replacement. You pay $2 to play and could profit $100,000 if you match all 5 numbers in order (you get your $2 back plus $100,000). Over the long term, what is your expected profit of playing the game?

To do this problem, set up an expected value table for the amount of money you can profit.

Let X = the amount of money you profit. The values of x are not 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Since you are interested in your profit (or loss), the values of x are 100,000 dollars and -2 dollars.

To win, you must get all 5 numbers correct, in order. The probability of choosing one correct number is 1 10 because there are 10 numbers. You may choose a number more than once. The probability of choosing all 5 numbers correctly and in order is:

1 10 * 1 10 * 1 10 * 1 10 * 1 10 * = 1 * 10 -5 = 0.00001

Therefore, the probability of winning is 0.00001 and the probability of losing is

1 - 0.00001 = 0.99999

The expected value table is as follows.

Αdd the last column. -1.99998 + 1 = -0.99998
x P(x) x P(x)
Loss -2 0.99999 (-2)(0.99999)=-1.99998
Profit 100,000 0.00001 (100000)(0.00001)=1

Since -0.99998 is about -1 , you would, on the average, expect to lose approximately one dollar for each game you play. However, each time you play, you either lose $2 or profit $100,000. The $1 is theaverage or expected LOSS per game after playing this game over and over.

Suppose you play a game with a biased coin. You play each game by tossing the coin once. P(heads) = 2 3 and P(tails) = 1 3 . If you toss a head, you pay $6. If you toss a tail, you win $10. If you play this game many times, will you come out ahead?

Define a random variable X .

X = amount of profit

Complete the following expected value table.

x ____ ____
WIN 10 1 3 ____
LOSE ____ ____ -12 3
x P(x) x P(x)
WIN 10 1 3 10 3
LOSE -6 2 3 -12 3

What is the expected value, μ ? Do you come out ahead?

Add the last column of the table. The expected value μ = -2 3 . You lose, on average, about 67 cents each time you play the game so you do not come out ahead.

Like data, probability distributions have standard deviations. To calculate the standard deviation ( σ ) of a probability distribution, find each deviation from its expected value, square it, multiply it by its probability, add the products, and take the square root . To understand how to do the calculation, look at the table for thenumber of days per week a men's soccer team plays soccer. To find the standard deviation, add the entries in the column labeled ( x μ ) 2 · P ( x ) and take the square root.

x P(x) x P(x) (x -μ) 2 P(x)
0 0.2 (0)(0.2) = 0 ( 0 - 1.1 ) 2 ( .2 ) = 0.242
1 0.5 (1)(0.5) = 0.5 ( 1 - 1.1 ) 2 ( .5 ) = 0.005
2 0.3 (2)(0.3) = 0.6 ( 2 - 1.1 ) 2 ( .3 ) = 0.243

Add the last column in the table. 0.242 + 0.005 + 0.243 = 0.490 . The standard deviation is the square root of 0.49 . σ = 0.49 = 0.7

Generally for probability distributions, we use a calculator or a computer to calculate μ and σ to reduce roundoff error. For some probability distributions, there are short-cut formulas that calculate μ and σ .

Questions & Answers

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Elementary statistics. OpenStax CNX. Dec 30, 2013 Download for free at http://cnx.org/content/col10966/1.4
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