Chapter 3, Section 11 Discrete Random Variables

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Chapter 3, Section 11Discrete Random Variables

• Tchebysheffs Theorem
• Chebyshevs Inequality

? John J Currano, 01/24/2009
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Tchebysheffs Theorem (or Chebyshevs Inequality)
is useful in approximating probabilities in cases
where we do not know the distribution of a random
variable and where it varies greatly from the
bell-shaped distribution where the empirical
rule applies (see Section 1.3 more in Chapter 7
next semester). Unfortunately, Tchebysheffs
Theorem gives limits on the probabilities which
are not great.
? John J Currano, 09/16/2007
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Note that while k must be positive, it need not
be an integer.
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Example. A blind will fit Myras bedrooms window
if its width is between 41.5 and 42.5 inches.
Myra buys a blind from a store that has 30 such
blinds. What can be said about the probability
that it fits her window if the average of the
widths of the blinds is 42 inches with standard
deviation 0.25 inch?
Solution. Let Y be the width of the blind that
Myra purchased. Then Y has mean µ 42 and
standard deviation?? 0.25. We know from
Tchebysheffs inequality that
Since 41.5 42 ? 2 (0.25) µ ? 2? and 42.5
42 2 (0.25) µ ? 2?, we can use Tchebysheffs
inequality with k 2. Thus
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There is another example on the class
website Example. Use Tchebysheffs Theorem to
determine how many times a fair coin must be
tossed in order for the probability to be at
least 0.90 that between 40 and 60 of the tosses
will be heads. It is shown that using
Tchebysheffs Theorem, n 250 tosses will
guarantee that between 40 and 60 of the tosses
will be heads with probability at least 0.90.
However, using the normal approximation to the
binomial, which will be discussed next semester
in Chapter 7, it can be shown that about 70
tosses will suffice.