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Math 145
February 20, 2012
Random Variable
– 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.
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}
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)
Rolling a Pair of Dice
Random variable: X3 = Total no. of dots
2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
5
6
7
8
9
10
6
7
8
9
10
11
7
8
9
10
11
12
Rolling a Pair of Dice
X4 = (positive) difference in the no. of dots
0
1
2
3
4
5
1
0
1
2
3
4
2
1
0
1
2
3
3
2
1
0
1
2
4
3
2
1
0
1
5
4
3
2
1
0
Rolling a Pair of Dice
X5 = Higher of the two.
1
2
3
4
5
6
2
2
3
4
5
6
3
3
3
4
5
6
4
4
4
4
5
6
5
5
5
5
5
6
6
6
6
6
6
6
More Examples
Survey:
Random variables :
X6 = Age in years.
X7 = Gender {1=male, 0=female}.
X8 = Height.
Medical Studies:
Random variables :
X9 = Blood Pressure.
X10 = {1=smoker, 0=non-smoker}.
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
Prob.
0
1/8
1
3/8
2
3/8
3
1/8
Probability Histogram
Tossing a coin 3 times:
Random variable : X1 = The number of heads.
X
0
1
2
3
Prob.
1/8
3/8
3/8
1/8
1/2
3/8
1/4
Prob
1/8
0
0
1
2
3
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)
Rolling a Pair of Dice
Random variable: X3 = Total no. of dots
x
P
2
2
3
4
5
3
4
5
6
4
5
6
7
5
6
7
8
6
7
8
9
7
8
9
10
6
7
7
8
8
9
9
10
10
11
11
12
3
4
5
6
7
8
9
10
11
12
1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
Rolling a Pair of Dice
Random variable: X3 = Total no. of dots
x
P
2
3
4
5
6
7
8
9
10
11
12
1/36 2/36 3/36 4/36 5/36 6/36 5/36 4/36 3/36 2/36 1/36
1/4
1/6
5/36
1/9
1/8
Prob
5/36
1/9
1/12
1/12
1/18
1/18
1/36
1/36
0
2
3
1. Pr(X3<5)=
4
5
6
7
8
9
10
11
2. Pr(3<X3<12)=
12
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,
1. Every pi is a between 0 and 1.
2. p1 + p2 +…+ pk = 1.
Mean of a Discrete R.V.
The probability distribution of X
x
x1
x2
x3
…
xk
Prob
p1
p2
p3
…
pk
1. Mean () = E(X) = x1p1+x2p2+…+ xkpk
2. Variance (2) = V(X) = (x1-)2p1 + (x2-)2p2
+ …+ (xk-) 2pk .
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.
Continuous Distributions
1.
2.
3.
4.
5.
6.
Normal Distribution
Uniform Distribution
Chi-squared Distribution
T-Distribution
F-Distribution
Gamma Distribution
Thank you!