MATH 1107Elementary Statistics

Lecture 8

Random Variables

Math 1107 – Random Variables• In Class Exercise with Dice

Math 1107 – Random Variables• Probability Distribution for a Dice Roll:

X P(X)

1 1/6

2 1/6

3 1/6

4 1/6

5 1/6

6 1/6

Math 1107 – Random Variables• What is the mean?• What is the standard deviation?• Are these outcomes discrete or

continuous?• What is the probability of each outcome?• What is the probability distribution?• Are these outcomes dependent upon the

previous roll?

Math 1107 – Random VariablesThere are two things that must be true for every probability distribution:

1. The Summation of all probabilities must equal 1;

2. Every individual probability must be between 0 and 1.

Math 1107 – Random Variables

Important Formulas and applications:1. μ = Σ [x*P(x)]

From the Probability table for Dice:

μ = (1*1/6)+(2*1/6)+(3*1/6)+(4*1/6)+(5*1/6)+(6*1/6) =3.5

Math 1107 – Random Variables

Important Formulas and applications:2. σ2 = Σ [(x- μ)2 * P(x)]

From the Probability table for Dice:

σ2 = ((1-3.5)2*.1667)+((2-3.5) )2*.1667)…

+((6-3.5) )2*.1667) = 2.92

Math 1107 – Random Variables

Important Formulas and applications:3. σ = SQRT(Σ [(x- μ)2 * P(x)])

From the Probability table for Dice:

σ = SQRT( 2.92) = 1.71

Math 1107 – Random Variables

An important note on rounding…keep your numbers in your calculator/computer and only round at the end!

Math 1107 – Random Variables

What is an unusual event? When should we be suspect of results?

Ultimately, you need to KNOW YOUR DATA to determine what makes sense or not. But here is a rule of thumb –

If an event is more than 2 standard deviations away from the mean, it is “unusual”:

μ + or - 2σ

Math 1107 – Random Variables

Expected Values:

Knowing something about the distribution of events, enables us to create an expected value. This is calculated as:

E(x) =

But note that the “expected value” may not be logical when dealing with discrete numbers.

Σ(x* P(x))

Math 1107 – Random Variables

Would you want to play a game where you had a 75% chance of winning $5 and a 25% chance of losing $10?

The expected value of this game is:

(.75*5)+(.25*-10) = $1.25

Math 1107 – Random Variables

Would you be willing to play a lottery for $1 where the chances of winning $100K were 1/1000?

The expected value of this game is:

(1/1000* 100,000)+(999/1000*-1) = $99

Would you be willing to play a lottery for $1 where the chances of winning $10M were 1/15,625,000,000? (this is 6 number 1-50)

The expected value is:(1/15,625,000,000)*10,000,000)+(15,624,999,999/15,625,000,000*-1) = -0.99936