Sections 4.1 lecture notes
Math331, Fall 2008
Instructor: David Anderson
Section 4.1: Random Variables
Example 1. Consider rolling a fair die twice.
S={(i, j) : i, j ∈ {1,...,6}}.
Suppose we are interested in computing the sum. Let Xbe the sum. Then X∈ {2,3,...,12}
is random as it depends upon the outcome of the experiment. It is a random variable. We
can compute probabilities associated with X.
P(X= 2) = P{(1,1)}= 1/36
P(X= 3) = P{(1,2),(2,1)}= 2/36
P(X= 4) = P{(1,3),(2,2),(1,3)}= 3/36.
Can write succinctly
Sum, i2 3 4 5 6 ... 12
P(X=i) 1/36 2/36 3/36 4/36 5/36 ··· 1/36
Loose Definition: Let Sbe a sample space. Then a function X:S→Ris a random
variable.
Technical Definition in book: Let Sbe a sample space. Then X:S→Ris a random
variable if for each interval I⊂R,{s∈S:X(s)∈I}is an event. (has to do with sets
with no probability)
Terminology: Instead of {s∈S:X(s)∈I}, we usually write {X∈I}or X∈I.
Example 2. Consider a bin with 5 white and 4 red chips. Let Sbe the sample space
associated with selecting three chips, with replacement, from the bin. Let Xbe the number
of white chips chosen.
So, S={(a, b, c) : a, b, c ∈ {W, R}}. We have X(R, R, R) = 0, X(W, R, R) =
X(R, W, R) = X(R, R, W ) = 1, X(W, W, R) = X(W, R, W ) = X(R, W, W ) = 2, and
X(W, W, W) = 3. We want probabilities.
P(X= 0) = P(R, R, R) = (4/9)3
P(X= 1) = P((W, R, R)∪(R, W, R)∪(R, R, W ))
=P(W, R, R) + P(R, W, R) + P(R, R, W ) = 3 ∗(5/9)(4/9)2
P(X= 2) = 3(5/9)2(4/9)
P(X= 3) = (5/9)3.
Note: (4/9)3+ 3 ∗(5/9)(4/9)2+ 3(5/9)2(4/9) + (5/9)3= 1.
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