A Random Variable X with Pdf G(X) = Λα Γ(Α) X ≥ 0 Has Gamma

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. 0Properties 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. The density function of U ∼ χ21 is x−1/2 e−x/2 <ul style="display: flex;"><li style="flex:1">,</li><li style="flex:1">x > 0. </li></ul>√fU (x) = 2π Remark: A χ21 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. Using Remark 3. and the above remark, a χ2k rv. follows gamma distribution with parameters α = k/2 and λ = 1/2. Thus the density function of V ∼ χ2k is: 1fV (x) = xk/2−1e−x/2 <ul style="display: flex;"><li style="flex:1">,</li><li style="flex:1">x > 0 </li></ul>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. Definition: If Z ∼ N(0, 1) and U ∼ χ2n and Z and U are independent, then the distribution of ZqU/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 <ul style="display: flex;"><li style="flex:1"> </li><li style="flex:1">!</li></ul>−(n+1)/2 Γ[(n + 1)/2] √t2 n<ul style="display: flex;"><li style="flex:1">f(t) = </li><li style="flex:1">1 + </li></ul>.nπ Γ(n/2) 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: (i) If T ∼ tn, then T2 ∼ F1,n. (ii) If X ∼ Fn,m, then X−1 ∼ Fm,n. 2

A Random Variable X with Pdf G(X) = Λα Γ(Α) X ≥ 0 Has Gamma

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