1. Probability Distributions
Random Variable: Represents a numerical value associated with each outcome of a probability experiment:
Example: Flip a coin three times:
Let x be number of heads, then
x = 0, 1, 2, 3
Pictures:

2. Types of Data
Quantitative variables are either discrete or continuous.
Discrete: Counts.
Must “jump”
from one data point
to the next.
Continuous: Measures:
Can be made more and
more precise.

3. Probability Distributions
A Discrete Probability Distribution lists each value for the random variable and its associated probability. All probabilities sum to 1.
Example: Toss a coin three times
X = number of heads
Sample Space:
HHH, HTT, THT, TTH
THH, HTH, HHT, TTT X
P(x)
1/8 1
3/8 2 3
total

4. Probability Distributions
Example: Toss a coin three times X
P(x)
1/8 1
3/8 2 3
total
X = number of heads
Sample Space:
HHH, HTT, THT, TTH
THH, HTH, HHT, TTT
3/8
3/8
1/8
1/8

5. Probability Distributions
Example: Is this a probability distribution?
No, because
<>1 X
P(x) 5
.28 6
.21 7
.43 8
.15
total
1.07

6. Probability Distributions
Example: Is this a probability distribution?
Yes, because =1 X
P(x) 5
.28 6
.21 7
.43 8
.08
total 1

7. Probability Distributions
Example: Make this a a probability distribution.
If the probability for 7 is .36, then all probabilities add to one. X
P(x) 5
.28 6
.21 7 ? 8
.15
total 1

8. Probability Distributions
Find the mean of a probability distribution:
Calculate the same as the mean of a frequency distribution X
P(x)
X*p(x) 5
.28
1.4 6
.21
1.26 7
.36
2.52 8
.15
1.2
total 1
E(x) = 6.38

9. Probability Distributions
Find the standard deviation of a probability distribution:
Var(x) = E(x^2) – E(x)^2 = 41.8 – = 1.096
SD = sqrt(1.096) = 1.05
Calculate the same as the standard deviation of a frequency distribution X
P(x)
X*p(x)
X^2*p(x) 5
.28
1.4 7 6
.21
1.26
7.56
.36
2.52
17.64 8
.15
1.2
9.6
total 1
E(x) = 6.38
E(x^2) =41.8

10. Probability Distributions
What is the expected value and SD for rolling a pair of dice? X
P(x)
X*P(x)
X^2*p(x) 2
1/36
2/36
4/36 3
6/36
18/36 4
3/36
12/36
48/36 5
20/36
100/36 6
5/36
30/36
180/36 7
42/36
294/36 8
40/36
320/36 9
36/36
324/36 10
300/36 11
22/36
242/36 12
144/36
Total
36/36 = 1
256/36=7
1974/36
=54.833
Expected value =7
variance
= = 5.833
Standard deviation
= sqrt(5.833)
=2.42

11. Probability Distributions
Probability function:
Is P(x) =(x+2)/14 for x = 0, 1, 2, 3 a probability function?
yes! All probabilites<1 and sum to 1 X
P(x)
2/14 1
3/14 2
4/14 3
5/14

12. Probability Distributions
Probability function:
Is P(x) =(x+2)/10 for x = 0, 1, 2, 3 a probability function?
No! All probabilities do not sum to 1 X
P(x)
2/10 1
3/10 2
4/10 3
5/10
14/10

13. Probability Distributions
You are playing Deal or No Deal. There are three cases left plus your case the amounts left are $1; $50; $10,000; and $500,000.
What is the expected value?
b) The banker makes you an offer of $100,000. Is this more or less than the expected value? X
P(x) 1
.25 50
10,000
500,000
c) If you could play and infinite number of games, the best strategy would be to never take an offer less than the expected value and to always take an offer more than the expected value. Why might this strategy differ for a single game?

14. Probability Distributions
You are playing Deal or No Deal. There are three cases left plus your case the amounts left are $1; $50; $10,000; and $500,000.
What is the expected value?
b) The banker makes you an offer of $100,000. Is this more or less than the expected value? X
P(x)
X*P(x) 1
.25 50
50/4
10,000
10000/4
500,000
500000/4
c) If you could play and infinite number of games, the best strategy would be to never take an offer less than the expected value and to always take an offer more than the expected value. Why might this strategy differ for a single game?