Transcript Random Variables

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
•
•
•
•
•
•
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
Examples for consideration…Are the following
probability distributions?
X
P(X)
X
P(X)
Green
.2
1
.167
Blue
.5
2
.334
Red
.1
3
.167
Purple
1.2
4
.334
5
.167
6
.334
Math 1107 – Random Variables
Important Formulas and applications:
1. μ = Σ [x*P(x)]
From the (correct) 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
Lets say that you are a teacher. Joe and Jimmy sit next to each
other in class. Here are their grades on the last 5 quizzes:
Quiz
Joe
Jimmy
1
92
51
2
95
50
3
98
48
4
91
49
5
93
93
Joe’s average = 94
Jimmy’s average (1st 4) = 50, with a std of 1.29. His
score on quiz 5 is 33 std from the mean.
Math 1107 – Random Variables
How do power companies detect people growing illegal plants in
their homes?
How do Pit Bosses determine who to throw out of a casino?
How do regulatory agencies determine which athletes to test for
illegal substances?
The rate of autism is approximately 1 in 166. What is the
probability of having a child with autism? Now, what is the
probability of having 2 children with autism? What would you
conclude?
Math 1107 – Random Variables
Example from Page 190:
Lets say that we have determined that the probability of having
a girl follows the distribution on page 183. A process called
“MICROSORT” enabled 13 out of 14 couples to have a girl, which
was their preference. Is this process successful?
According to the table on page 183, the probability of getting 13
out of 14 girls is .001. This suggests that MICROSORT’s
performance is successful.
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))
Note that this is a similar calculation to the
mean of a distribution.
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
Assuming that you are “risk neutral” you would be
willing to pay no more than $1.25 to play this game.
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
Math 1107 – Random Variables
When you give a casino $5 for a bet on the number
10 in roulette, you have a 1/38 probability of winning
$175 and a 37/38 probability of losing $5. What is
your expected value?
(175*1/38)+(-5*37/38) = -$.26
Should you play?