Example:

Two-dice totals. There are 36 possible simple events when two dice are thrown but a gambler may only want to know the total.

 
In this case, X is the discrete random variable and takes on values:
 2, 3, 4, 5, ... ,12

A Summary Table of all values of X and Prob(X) is usually given:

Click on the variable for an example
2	0.028
3	0.056
4	0.083
5	0.111
6	0.139
7	0.167
8	0.139
9	0.111
10	0.083
11	0.056
12	0.028

P(X)= 1/36

There is one possible 2

P(X) = 4/36

There are fourpossible 9s

P(X) = 2/36

There are two possible 3s

P(X) = 3/36

There are three possible 4s

P(X)= 4/36

There are four possible 5s

P(X) = 5/36

There are five possible 6s

P(X) = 6/36

There are six possible 7s

P(X) = 5/36

There are five possible 8s

P(X) = 3/36

There are three possible 10s

P(X) = 2/36

There are two possible 11s

P(X)= 1/36

There is one possible 12

Following the above values for X, we would need Prob(0) , Prob(1) , Prob(2), ... to be assigned proper values.

Requirements for a probability distribution:

1. Every probability that is given , say Prob(X),      must be between 0% and 100%.

    Again, 0% < Prob(X) < 100% or 0 < P(X) < 1.00

2. The sum of all these probabilities must be 1.00      or 100%.

      

Definition:

Typically we use X as the random variable.

A discrete random variable has either a finite number of values or a countable number of values.

For example, we may use
X = 0 , 1 , 2 , 3 , 4 , ....
for the discrete random variable.

A probability distribution gives the probability for each value of the random variable.

Example:

We are given random variable X values of 0,1,2,3,4,5,6,7.

Probabilities are given :
P(0)= 0.09 P(1)= 0.16 P(2)= 0.21 P(3)=0.19 P(4)=0.14
P(5)= 0.08 P(6) = 0.08 P(7)=0.05

Question:   

Make a summary table of X and P(X) values. Make a histogram.

0 0.09 1 0.16 2 0.21 3 0.19 4 0.14 5 0.08 6 0.08 7 0.05

Probability distributions are a basis for much of Probability and Statistics.

For example, if you could see a probability distribution of Temperature for the month of January then you could plan which temperatures you are most likely to encounter.

Some probability distributions have formulas:

Example:

Prob(X) = 2^-x
X =1,2,3,4,5,6,.....

Question:

Prove this is a probability distribution.
Make a histogram.

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Toss 1
Toss 2
Toss 3
Toss 4

 

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Toss 1
Toss 2
Toss 3
Toss 4

 

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Toss 1
Toss 2
Toss 3
Toss 4

 

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Toss 1
Toss 2
Toss 3
Toss 4

 

	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
Toss 1
Toss 2
Toss 3
Toss 4

 

Question :
What is the probability of
4 "Heads" out of four tosses ?
How about 3 "Heads" out of 4?
Tossing 2 "Heads" out of4 ?
Tossing 1 "Heads"out of 4 ?
Tossing 0 "Heads" out of4 ?We need a random variable X which is properly defined:

X = number of "Heads" out of 4 .
Clearly values of X are 0 , 1 , 2 ,3 , 4

0	1/16	6.25%
1	4/16	25.0%
2	6/16	37.5%
3	4/16	25.0%
4	1/16	6.25%