2 3 the maclaurin series expansion of a function f is

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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 discrete-type) 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 discrete-type 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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