DISTRIBUTIONS DERIVED FROM THE NORMAL DISTRIBUTION Definition: A random variable X with pdf λα g(x) = xα−1e−λx x ≥ 0 Γ(α) has gamma distribution with parameters α > 0 and λ > 0. The gamma function Γ(x), is defined as Z ∞ Γ(x) = ux−1e−udu. 0 Properties of the Gamma Function: (i) Γ(x + 1) = xΓ(x) (ii) Γ(n + 1) = n! √ (iii) Γ(1/2) = π. Remarks: 1. Notice that an exponential rv with parameter 1/θ = λ is a special case of a gamma rv. with parameters α = 1 and λ. 2. The sum of n independent identically distributed (iid) exponential rv, with parameter λ has a gamma distribution, with parameters n and λ. 3. The sum of n iid gamma rv with parameters α and λ has gamma distribution with parameters nα and λ. Definition: If Z is a standard normal rv, the distribution of U = Z2 called the chi-square distribution with 1 degree of freedom. 2 The density function of U ∼ χ1 is −1/2 x −x/2 fU (x) = √ e , x > 0. 2π 2 Remark: A χ1 random variable has the same density as a random variable with gamma distribution, with parameters α = 1/2 and λ = 1/2. Definition: If U1,U2,...,Uk are independent chi-square rv-s with 1 degree of freedom, the distribution of V = U1 + U2 + ... + Uk is called the chi-square distribution with k degrees of freedom. 2 Using Remark 3. and the above remark, a χk rv. follows gamma distribution with parameters 2 α = k/2 and λ = 1/2. Thus the density function of V ∼ χk is: 1 f (x) = xk/2−1e−x/2, x > 0 V 2k/2Γ(k/2) 1 Proposition: If V has a chi-square distribution with k degree of freedom, then E(V ) = k, Var(V ) = 2k. 2 Definition: If Z ∼ N (0, 1) and U ∼ χn and Z and U are independent, then the distribution of Z q U/n is called the t distribution with n degrees of freedom. Proposition: The density function of the t distribution with n degrees of freedom is Γ[(n + 1)/2] t2 !−(n+1)/2 f(t) = √ 1 + . nπ Γ(n/2) n Remarks: For the above density f(t) = f(−t), so the t density is symmetric about zero. As the number of degrees of freedom approaches infinity, the t distribution tends to be standard normal. Definition: Let U and V be independent chi-square variables with m and n degrees of freedom, respectively. The distribution of U/m W = V/n is called F distribution with m and n degrees of freedom and is denoted by Fm,n. Remarks: 2 (i) If T ∼ tn, then T ∼ F1,n. −1 (ii) If X ∼ Fn,m, then X ∼ Fm,n. 2.