Unformatted text preview: ust be TAIL) = P (TTTT ....H )
(tosses are independent) = (1 − p )k −1 p ,
M. Chen ([email protected]) ENGG2430C lecture 5 k = 1, 2, ....
11 / 21 Computing a PMF pX (x )
Collect all possible outcomes ω ∈ Ω, such that X (ω ) = x :
{ω : ω ∈ Ω, X (ω ) = x }
pX (x ) = P ({ω : ω ∈ Ω, X (ω ) = x }). Repeat for all x ∈ R .
Example: two independent throws of a fair die.
Deﬁne X and Y as the outcomes of the ﬁrst and second throw,
respectively
9
Deﬁne Z = min(X ;Y ). pZ (2) = 36 M. Chen ([email protected]) ENGG2430C lecture 5 12 / 21 Binomial Random Variable Experiment: ﬂip a coin Independently for n times. Deﬁne
Bernoulli/indicator r.v.s Xi , (i = 1, 2, . . . , n) be the outcome of the
i th ﬂip:
1, the ith toss gives HEAD;
Xi =
0, the ith toss gives TAIL.
The PMF is pXi (1) = p and pXi (0) = 1 − p . Let Y = ∑n=1 Xi . Then Y is a Binomial random variable. Its PMF is
i
given by
n pY (k ) = P (Y = k ) = P ∑ Xi = k i =1 = (n ) p k (1 − p )n−k .
k
M. Chen ([email protected]) ENGG2430C lecture 5 13 / 21 Service Facility Design n: the number of customers
p : probability that a customer requires service
s : number of service lines
X : a r.v. denoting the number of service requests
Goal: Design a system that “noservice" probability is no more than ε :
P (X > s ) = PX (s + 1) + · · · + pX (n)
n
n
=
∑ k p k (1 − p )n−k .
k =s +1
The larger the s , the smaller the “noservice" probability is. However,
it costs more to run the service.
Given ε , the optimal s ∗ (ε ) = min(s ∈ N : P (X > s ) ≤ ε ).
M. Chen ([email protected]) ENGG2430C lecture 5 14 / 21 Expectation The socalled ﬁrstorder statistics about random outcomes:
E [X ] = ∑ xpX (x ). (A weighted sum of outcomes x ’s.)
x Interpretations:
The center of gravity of the probability mass
The average of repeated experiments Example: a sample space Ω = {0, 1, . . . , n} with a uniform probability
law (From MIT open course 6.041 / 6.431 lecture notes.) E [X ] = 0 ×
M. Chen ([email protected]) 1
1
n
n
+1×
+···+
=
n+1
n+1
n+1 2
ENGG2430C lecture 5 15 / 21 Examples Calculate the expectation of a Bernoulli random...
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 Spring '14

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