Download The normal distribution A continuous random variable X is said to be

Probability and Statistics
Grinshpan
The normal distribution
A continuous random variable X is said to be normal or Gaussian if its density function is
2
2
1
f (x) = √
e−(x−µ) /2σ ,
σ 2π
where µ and σ > 0 are constants. The fact that f (x) is a probability density function is supported by
the formula
∫ ∞
√
2
e−x dx = π.
−∞
One has E[X] = µ, Var(X) = σ 2 , and writes X ∼ N (µ, σ 2 ). The standardized normal variable
Z = (X − µ)/σ has mean 0 and variance 1. The cumulative distribution function of Z is
∫ x
2
1
Φ(x) = P (Z < x) = √
e−s /2 ds.
2π −∞
One has
∫ b
2
1
e−x /2 dx = Φ(b) − Φ(a).
P (a < Z < b) = √
2π a
Due to even symmetry of the density function, Φ(x) + Φ(−x) = 1. There exists no expression for Φ(x) in
terms of elementary functions. However, numerically, its values are available with good precision.
A variant of Φ(x) is the error function
∫ x
√
2
2
erf(x) = √
e−s ds = 2Φ( 2 x) − 1.
π 0
The probability that X ∼ N (µ, σ 2 ) falls within certain bounds is easy to write as a change in Φ :
P (a < X < b) = P ((a − µ)/σ < Z < (b − µ)/σ) = Φ((b − µ)/σ) − Φ((a − µ)/σ).
In particular, P (aσ < X − µ < bσ) = Φ(b) − Φ(a). The three-sigma rule states that practically all values
of a normal random variable fall within 3 standard deviations of its mean. In fact,
P (|X − µ| < σ) = Φ(1) − Φ(−1) = 2Φ(1) − 1 ≈ 0.68
P (|X − µ| < 2σ) = 2Φ(2) − 1 ≈ 0.95
P (|X − µ| < 3σ) = 2Φ(3) − 1 ≈ 0.997.
2
The moment-generating function of Z is M (t) = et /2 . This can be shown by a direct calculation:
∫ ∞
∫ ∞
∫ ∞
2
2
2
2
2
2
1
1
1
M (t) = √
etx e−x /2 dx = √
et /2 e−(x−t) /2 dx = et /2 √
e−s /2 ds = et /2 .
2π −∞
2π −∞
2π −∞
The moments of Z are thus
{
1 · 3 · · · (k − 1), k even,
k
E[Z ] =
0,
k odd.
2
If Y is a random variable whose moment-generating function agrees with et /2 in an open interval
containing zero, then Y ∼ N (0, 1). In fact, if the moment-generating function of Y is sufficiently close to
2
et /2 , then Y is nearly the standard normal.