Excursions in Modern Mathematics, 7e: 16.5 - 1Copyright © 2010 Pearson Education, Inc.

16 Mathematics of Managing Risks

• Weighted Average

• Expected Value

Excursions in Modern Mathematics, 7e: 16.5 - 2Copyright © 2010 Pearson Education, Inc.

The weighted average (or weighted mean) of a set of N numbers

each of which is assigned a weight

where is:

Weighted Average or Weighted Mean

Nvvv ,...,, 21

Nwww ,...,, 21

121 Nwww

NN wvwvwv 2211

Excursions in Modern Mathematics, 7e: 16.5 - 3Copyright © 2010 Pearson Education, Inc.

Examples

If homework/quiz average is weighted 20%, 2 exams are weighted 25% each, and final exam is weighted 30% and a student makes homework/quiz average 87, exam scores of 80 and 92, and final exam score 85. Compute the weighted average.

Excursions in Modern Mathematics, 7e: 16.5 - 4Copyright © 2010 Pearson Education, Inc.

Examples

9.85)85(30.0)92(25.0)80(25.0)87(20.0

The weighted average is

Excursions in Modern Mathematics, 7e: 16.5 - 5Copyright © 2010 Pearson Education, Inc.

Random Variable

A random variable is a letter (X) that denotes a single numerical value which is observed when performing a random experiment.

Excursions in Modern Mathematics, 7e: 16.5 - 6Copyright © 2010 Pearson Education, Inc.

• Toss a coin 3 times and count the number of heads. Denote the total number of heads by the random variable X.

• A basketball player shoots two consecutive free throws. Denote the total number of points scored by the random variable X.

Examples of Random Variable

Excursions in Modern Mathematics, 7e: 16.5 - 7Copyright © 2010 Pearson Education, Inc.

Probability Distribution

A probability distribution for a random variable X gives the probability for any value of X. (Note: this is similar to a probability assignment for a sample space)

Example:

Toss a coin 3 times and count the number of heads. Denote the total number of heads by the random variable X. What is the probability distribution for X?

Excursions in Modern Mathematics, 7e: 16.5 - 8Copyright © 2010 Pearson Education, Inc.

Probability Distribution

X 0 1 2 3

P(X) 1/8 = 0.125 3/8 = 0.375 3/8 = 0.375 1/8 = 0.125

Excursions in Modern Mathematics, 7e: 16.5 - 9Copyright © 2010 Pearson Education, Inc.

The expected value (E) of a random variable X which has N possible outcomes

each of which is assigned a probability

where is:

Nxxx ,...,, 21

Nppp ,...,, 21

121 Nppp

NN pxpxpxE 2211

Expected Value of a Random Variable

Excursions in Modern Mathematics, 7e: 16.5 - 10Copyright © 2010 Pearson Education, Inc.

Expected Value of a Random Variable

• The formula for the expected value is similar to a weighted average formula.

• The expected value of a random variable X gives the approximate value of X that would result after repeating the random experiment many, many times.

Excursions in Modern Mathematics, 7e: 16.5 - 11Copyright © 2010 Pearson Education, Inc.

Example

Toss a coin 3 times and count the number of heads. Denote the total number of heads by the random variable X. What is the expected value of X? (Use the probability distribution in the previous example.)

Excursions in Modern Mathematics, 7e: 16.5 - 12Copyright © 2010 Pearson Education, Inc.

Example

5.1)3(125.0)2(375.0)1(375.0)0(125.0 E

X 0 1 2 3

P(X) 1/8 = 0.125 3/8 = 0.375 3/8 = 0.375 1/8 = 0.125

That is, we expect there will be 1.5 heads in three tosses (that is, we expect that 50% of the tosses would result inheads).

Excursions in Modern Mathematics, 7e: 16.5 - 13Copyright © 2010 Pearson Education, Inc.

Example

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Excursions in Modern Mathematics, 7e: 16.5 - 14Copyright © 2010 Pearson Education, Inc.

Example

• X is a random variable that represents the net gain (or loss) of your bet.

• Probability distribution of X is (assuming each guess equally likely):

X -$1 $36

P(X) 37/38 1/38

03.038

1)36(

38

1)1(

38

37E

Excursions in Modern Mathematics, 7e: 16.5 - 15Copyright © 2010 Pearson Education, Inc.

Example

The negative indicates that if the random experiment were repeated many times, there would be a net loss of about $0.03 (house wins).