10F (Probability)

n

and rate one. Direct integration gives, for

t < 1

M_{X} (t) = \frac{1}{( n - 1 )!} \int_{0}^{\infty} x^{n - 1} e^{- (1 - t) x} d x = (1 - t)^{- n} .

b

Differentiating the moment-generating function at zero gives

E X = M_{X}^{'} (0) = n,

var X = M_{X}^{''} (0) - M_{X}^{'} (0)^{2} = n (n + 1) - n^{2} = n .

An Erlang

(n, 1)

variable is a sum of

n

independent rate-one exponential variables, each with mean and variance one. The central limit theorem therefore gives

\frac{X - n}{n} \Rightarrow N (0, 1) .

Consequently, for every fixed

q \geq 0

\int_{0}^{n + q n} \frac{x ^{n - 1} e ^{- x}}{( n - 1 )!} d x = P (\frac{X - n}{n} \leq q) ⟶ Φ (q) .

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