Transcript Lectures_8

MATH 1107
Elementary 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
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•
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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 Variables
There 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) =
Σ(x* P(x))
But note that the “expected value” may not be
logical when dealing with discrete numbers.
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 150)
The expected value is:
(1/15,625,000,000)*10,000,000)+(15,624,999,999/15,625,000,
000*-1) = -0.99936