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By G. Jameson

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Of course, not all have expected value 0 and variance 1. 6 A random variable X is said to have the distribution N(µ, σ2 ) if X has pdf fX given by 1 x−µ 2 1 fX (x) = √ e− 2 ( σ ) , 2πσ where µ ∈ R and σ > 0 are fixed. R EMARK: If σ = 1, then fX (x) = f(x − µ), where f is the pdf of the the N(0, 1) distribution. So the graph of fX (x) in this case is just that of f(x) moved µ units to the right. So if X has distribution N(µ, σ2 ) then X has expected value µ and variance σ2 . For fixed µ and σ, the graph of fX (x) is a “bell-shaped” curve, in which the value of fX (x) is negligible for X more than about 3σ away from µ.

N OTES: 1. If f is a pdf for X, then ∞ f(t)dt = 1. −∞ (This follows from limx→∞ FX (x) = 1). 2. If FX is differentiable for some random variable X, then its derivative FX is a pdf for X (this follows from the Fundamental Theorem of Calculus). 3. Suppose a, b ∈ R, a < b. Then P(a X b) = P(X b) − P(X = FX (b) − FX (a) b = a f(f)dt − ∞ b = a) f(t)dt −∞ f(t)dt. a b P(a X b) = f(t)dt. 7 Let X be a continuous random variable with pdf f. Then the expectation of X is defined by ∞ E(X) = tf(t)dt. 4, show that E(X) = π.

X: Negative Binomial Random Variable with parameters r and p X is the number of trials to get the rth success. 1 A random variable is called continuous if it can take values anywhere in an interval of the real numbers. Continuous random variables do not have probability functions in the sense discussed in Chapter 2. To discuss probability theory for continuous random variables, we need the idea of a cumulative distribution function. Let S be a sample space and let X : S −→ R be a random variable on S.

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