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Unformatted text preview: of the form Y ∈ B. In particular, P {X = i, Y = j } =
pX (i)pY (j ).
More generally, random variables (not necessarily discretetype) X1 , X2 , . . . , Xn are mutually
independent if any set of events of the form {X1 ∈ A1 }, {X2 ∈ A2 }, . . . , {Xn ∈ An } are mutually
independent. Independence of random variables is discussed in more detail in Section 4.4. 2.4.3 Bernoulli distribution Some distributions arise so frequently that they have names. Two such distributions are discussed
in this section: the Bernoulli and binomial distributions. The geometric and Poisson distributions
are two other important discretetype distributions with names, and they are introduced in later
sections.
A random variable X is said to have the Bernoulli distribution with parameter p, where 0 ≤ p ≤
1, if P {X = 1} = p and P {X = 0} = 1 − p. Note that E [X ] = p. Since X = X 2 , E [X 2 ] = E [X ] = p.
So Var(X ) = E [X 2 ] − E [X ]2 = p − p2 = p(1 − p). The variance is plotted as a functio...
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 Spring '08
 Zahrn
 The Land

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