Week 08-Continuous random variable and MGF

Gamma Distribution

A random variable X has a gamma(α, β) distribution if and only if it has a pdf
f (x) \=1/βαΓ(α) x^α−1 exp{−x/β}, x \> 0, α \> 0, β \> 0

α is called the shape parameter
and β is called the scale
parameter. In some textbook,
they use λ \= 1/β (the rate
parameter).

Γ(α) is the gamma function of α.

properties of gamma function

- For any α \> 1, Γ(α + 1) \= αΓ(α)
- For any positive integer n, Γ(n) \= (n − 1)!
- Γ(1) \= 1 and Γ(1/2) \=√π

χ2 distribution

The sampling or probability distribution for chi-square tests.

fX (x; k) \= 1/ 2k/2Γ(k/2)^x(k/2)−1 e−x/2, x ≥ 0

k is the degrees of freedom of X

Two special cases of Gamma distribution

when the α and β take two special set of values, it leads to the chi-squared distribution and exponential distribution

Exponential distribution

random variable, X, is Exponential(λ) distributed if it has the pdf
f (x) \= λe^−λx , x \> 0, λ \> 0
- is a 1-parameter sub-family of the gamma
family by setting α \= 1 and β \= 1/λ.
- λ is called the the rate parameter which is the reciprocal of the scale
parameter.

memoryless property

For an exponential random variable X, the memoryless property is the statement that knowledge of what has occurred in the past has no effect on future probabilities. This means that the probability that X exceeds x + k, given that it has exceeded x, is the same as the probability that X would exceed k if we had no knowledge about it. In symbols we say that P(X \> x + k|X \> x) \= P(X \> k).

Moment Generating Function (MGF)

M.X(t) \= E[e^(tX)]

Transformation: Y \= g(X)

FY (y) \=
FX (g^−1(y)), g() increasing
1 − FX (g^−1(y)), g() decreasing

Why model multiple variables

Many problems involve working simultaneously with two or more random
variables

bivariate distribution

Pairs of measures on two variables collected on the same subjects

Marginal distribution

Distribution of values of that variable among all individuals described by the table.