Lesson 4.2 Expected Value

Law of Large numbers

The mean of random variable X is μ. If we do an experiment over and over again, the average gets closer and closer to μ. This is known as the Law of Large Numbers.

Example: If we toss a fair coin 20,000 times and let X = the number of heads, then the mean of X, μ, is very close to 1/2.

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Expected Value

The expected value is known as "the long-term" average or mean, μ. This means that over the long term of doing an experiment over and over, you would expect this average every time you perform the experiment.

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

Example: In Section 4.1, a PDF table was created. It is shown below.

X = the number of patients out of two cured

P(a cure for one patient) = 5/6 and

P(no cure for one patient) = 1/6.

To find the expected value, add another column and label it X*P(X). Calculate each product (X*P(X)) and then add (sum) the products for the expected value.

X	P(X)	X*P(X)
0	P(X=0) = 1/36	(0)(1/36) = 0
1	P(X=1) = 10/36	(1)(10/36) = 10/36
2	P(X=2) = 25/36	(2)(25/36) = 50/36

Add the last column: 0 + 10/36 + 50/36 = 60/36 = 1.67. The expected value is one and two-thirds patients cured. That means that over the long-term, we would expect 1 2/3 patients out of 2 patients to be cured. Another way to look at it is as follows. If we surveyed 6 patients who had taken the drug, we would expect 5 to be cured.

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Expected Value Problem

Example

The following game problem is an example of an expected value problem. Close the window when you are finished viewing the example. You will return here.

Please continue to the next section of this lesson.

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Up » 4.1 Discrete Probability » 4.2 Expected Value » 4.3 Binomial Probability » 4.4 Poisson Probability

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