Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Example: Flip a fair coin twice. Let X(t) be the random variable that equals the
number of tails that appear when t is the outcome. Then
X(HH) = 0, X(HT) = X(TH) = 1, and X(TT) = 2.
Definition: The distribution of a random variable X on a sample space is the set
of pairs (r, p(X=r)) (r,X(S), where p(X=r) is the probability that X takes the
value r.
Note: A distribution is usually described by specifying p(X=r) (r,X(S).
Example: For our coin flip example above, each outcome has probability 0.25.
Hence,
p(X=0) = 0.25, p(X=1) = 0.5, and p(X=2) = 0.25.
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Definition: The expected value (or expectation) of the random variable X(s) in
the sample space S is E(X)= " p(s)X(s) .
s!S
n
i=1
Note: If S = {x i
} n i=1
, then E(X) = ! p(x i
)X(x i
).
Example: Roll a die. Let the random variable X take the valuess 1, 2, …, 6 with
n
!
$
probability 1/6 each. Then E = 1
' = 3.5. This is not really what you would
i=1"
# 6 %
&
like to see since the die does not a 3.5 face.
Theorem: If X is a random variable and p(X=r) is the probability that X=r so
that p(X=r) = " , then E(X) = " p(X=r)r .
p(s)
r!S,X(s)=r
r!X(S)
Proof: Suppose X is a random variable with range X(S). Let p(X=r) be the
probability that X takes the value r. Hence, p(X=r) is the sum of probabilities of
outcomes s such that X(s)=r Finally, E(X) = " p(X=r)r .
r!X(S)
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