MA/STAT416: Probability
Lecture Notes
Spring 2011
1
Chapter 4: Discrete Random Variables and Mass Functions
March 7, 2011
4.5
Uses of
μ
and
σ
as Summaries; Markov and Chebyshev’s Inequality
Theorem 1
(Chebyshev’s Inequality)
.
Suppose
E
(
X
) =
μ
and
V ar
(
X
) =
σ
2
are as
sumed to be finite. Let
k
be any positive number. Then,
P
(

X

μ
 ≥
kσ
)
≤
1
k
2
.
Example
2
.
Find the variance and standard deviation of the sum of the scores in rolling
a fair die twice.
•
How much of the probability distribution falls within two sigma from the mean?
Three sigma from the mean?
•
Compute the Chebyshev bound for the two sigma and the three sigma interval
around the mean. Compare the bound with the exact value of those probabilities.
Example
3
.
Find the variance and standard deviation of the number of aces in a hand
of 4 cards drawn randomly from a deck of 52 cards.
•
How much of the probability distribution falls within two sigma from the mean?
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 Spring '08
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 Probability, Standard Deviation

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