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Continuous random variables

Table 1.2 – Some important continuous probability distributions

Distribution Probability law f(x) E(X) Var(X)
1 Γ
Breit-Wigner 1 2 − −
π 1 Γ+(x−x 0 ) 2
4 [ 2 ]
Gaussian √ 1 exp − (x−µ) µ σ 2
σ 2π 2σ 2
( ) c−1
c
1
1
2
2
2
Weibull c x exp{−[x/b] } bΓ(1 + ) b (Γ(1 + ) − Γ (1 + ))
b b c c c
e
chi-squared 1 x (n/2)−1 −x/2 n 2n
2 n/2 Γ(n/2)
exponential λe −λx 1 1
λ λ 2
gamma λ (λx) r−1 −λx r r
e
Γ(r) λ λ 2
[ 2 ]
2
log-normal √ 1 1 exp − (ln x−µ) e µ+σ /2 (e σ 2 − 1)e 2µ+σ 2
σ 2π x 2σ 2
uniform 1 a+b (b−a) 2
b−a 2 12
The uniform distribution. The distribution of a continuous random variable whose density is
constant in some closed interval [a, b] and is equal to zero outside this interval is called a uniform
distribution:
{
1 , if x ∈ [a, b],
f(x) = b−a
0, if x /∈ [a, b]
+∞
∫ b 1
that is f(x) = c for x ∈ [a, b] but E(X) = cdx = 1, and hence cx = 1, c = . Instead of a
a b−a
−∞
closedinterval[a, b]itispossibletotake(a, b),or[a, b),(a, b]sincearandomvariableiscontinuous.
The mean and variance of the uniform distribution are given by
a + b (b − a) 2
E(X) = , Var(X) = .
2 12
The graph of the density f(x) has the form
y
1
b−a

0 a b x

Figure 6.4 – A density of uniform distribution

Auniformdistributionischaracteristicforthephaseofrandomoscillations. Inpracticewehave
to consider harmonic oscillations with random amplitude and phase. In such cases the phase is
often a random variable uniformly distributed over the period of oscillations.
Let us find the integral distribution function F(x).
∫ x
Since F(x) = f(t)dt, then
−∞
∫ x ∫ x
1. if x < a we have f(t)dt = 0dt = 0,
−∞ −∞
2. if a ≤ x ≤ b we obtain
∫ ∫ ∫ ∫ ∫
x a x a x dt x − a
f(t)dt = f(t)dt + f(t)dt = 0dt + = ,
b − a b − a
−∞ −∞ a −∞ a
3. if x > b we get

∫ ∫ ∫ ∫
x a b x
f(t)dt = f(t)dt + f(t)dt + f(t)dt =
−∞ −∞ a a

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