Expectation Farrokh Alemi Ph.D.

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Presentation transcript:

Expectation Farrokh Alemi Ph.D. In these slides we discuss the concept of Expectation. This brief presentation was organized by Farrokh Alemi and relies in part on Openbook on Statistics.

Random Variable Sample Space Event 1 Event 2 Event 3 X=x1 X=x2 X=x3 Random Variable X A random variable assigns numerical values to events within the sample space.

Probability of a Random Variable The random variable X is shown as capital X. The probability that the random variable X has the value X sub I is shown here.

Expected Value The expected value of a random variable is shown as the function E evaluated for the random variable X.

Expected Value The expected value of a discrete random variable is calculated summing product of the probability of the event times the value of the event.

Dental Service In this example, the charges for cleaning, root canal, crown and post are shown for a dental clinic. The probability of a patient needing any one of these services is also shown. What is the average revenue per patient to this dental service. Note that the probabilities of these events do not add up to 1 as some of these events may occur together, so one might have cleaning and a crown.

Dental Service The expected value can be calculated by summing the product of the value of the event times its probability.

Dental Service Expected contribution of cleaning is 72 dollars.

Dental Service Root canal contributes $150 to the cost of an average patient.

Dental Service The contribution of crown is $240.

Dental Service The expected cost is the sum of each of the contributions.

Dental Service The expected cost for an average patient is $500.

Revenues Suppose an optional home monitoring device is purchased by 80% of the patients. 55% buy just the device and 25% buy both the device and supplies needed for it. The expected cost can now be calculated by multiplying the dollar amount of sales with the probability of that sale.

Balance in Revenues Think of the expected value as a point that would balance the probability distribution of the events. For the example this occurs at $117.85.

Expected value of a random variable is the sum of the events weighted by probability of the events