1. Math 145

2. Random Variable Two Types:
– A random variable is a variable whose value is a numerical outcome of a random phenomenon.
– A random variable is a function or a rule that assigns a numerical value to each possible outcome of a statistical experiment.
Two Types:
1. Discrete Random Variable – A discrete random variable has a countable number of possible values (There is a gap between possible values).
2. Continuous Random Variable – A continuous random variable takes all values in an interval of numbers.

3. Examples Tossing a coin 3 times: Sample Space
= {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}.
Random variables :
X1 = The number of heads.
= {3, 2, 2, 2, 1, 1, 1, 0}
X2 = The number of tails.
= {0, 1, 1, 1, 2, 2, 2, 3}

4. Rolling a Pair of Dice Sample Space: (1, 1) (1, 2) (1, 3) (1, 4)
(1, 5)
(1, 6)
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)

5. Random variable: X3 = Total no. of dots
Rolling a Pair of Dice
Random variable: X3 = Total no. of dots 2 3 4 5 6 7 8 9 10 11 12

6. Rolling a Pair of Dice
X4 = (positive) difference in the no. of dots 1 2 3 4 5

7. Rolling a Pair of Dice
X5 = Higher of the two. 1 2 3 4 5 6

8. More Examples Survey: Medical Studies: Random variables :
X6 = Age in years.
X7 = Gender {1=male, 0=female}.
X8 = Height.
Medical Studies:
X9 = Blood Pressure.
X10 = {1=smoker, 0=non-smoker}.

9. Probability Distribution
Tossing a coin 3 times:
Sample Space
= {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}.
Random variable : X1 = The number of heads.
= {3, 2, 2, 2, 1, 1, 1, 0} x 1 2 3
Prob.
1/8
3/8

10. Probability Histogram
Tossing a coin 3 times:
Random variable : X1 = The number of heads. X 1 2 3
Prob.
1/8
3/8

11. Rolling a Pair of Dice Sample Space: (1, 1) (1, 2) (1, 3) (1, 4)
(1, 5)
(1, 6)
(2, 1)
(2, 2)
(2, 3)
(2, 4)
(2, 5)
(2, 6)
(3, 1)
(3, 2)
(3, 3)
(3, 4)
(3, 5)
(3, 6)
(4, 1)
(4, 2)
(4, 3)
(4, 4)
(4, 5)
(4, 6)
(5, 1)
(5, 2)
(5, 3)
(5, 4)
(5, 5)
(5, 6)
(6, 1)
(6, 2)
(6, 3)
(6, 4)
(6, 5)
(6, 6)

12. Random variable: X3 = Total no. of dots
Rolling a Pair of Dice
Random variable: X3 = Total no. of dots 2 3 4 5 6 7 8 9 10 11 12 x 2 3 4 5 6 7 8 9 10 11 12 P
1/36
2/36
3/36
4/36
5/36
6/36

13. Random variable: X3 = Total no. of dots
Rolling a Pair of Dice
Random variable: X3 = Total no. of dots x 2 3 4 5 6 7 8 9 10 11 12 P
1/36
2/36
3/36
4/36
5/36
6/36
1. Pr(X3<5)= 2. Pr(3<X3<12)=

14. Discrete Random Variable
A discrete random variable X has a countable number of possible values.
The probability distribution of X x x1 x2 x3 … xk
Prob p1 p2 p3 pk
where,
Every pi is a between 0 and 1.
p1 + p2 +…+ pk = 1.

15. Mean () = E(X) = x1p1+x2p2+…+ xkpk
Mean of a Discrete R.V.
The probability distribution of X x x1 x2 x3 … xk
Prob p1 p2 p3 pk
Mean () = E(X) = x1p1+x2p2+…+ xkpk
Variance (2) = V(X) = (x1-)2p1 + (x2-)2p2 + …+ (xk-) 2pk .

16. Continuous Random Variable
A continuous random variable X takes all values in an interval of numbers.
Examples: X11 = Amount of rain in October.
X12 = Amount of milk produced by a cow.
X13 = Useful life of a bulb.
X14 = Height of college students.
X15 = Average salary of UWL faculty.
The probability distribution of X is described by a density curve.
The probability of any event is the area under the density curve and above the values of X that make up the event.

17. Continuous Distributions
Normal Distribution
Uniform Distribution
Chi-squared Distribution
T-Distribution
F-Distribution
Gamma Distribution

18. Thank you!