(1) Uniform Distribution(2) Binomial Distribution(3) Poisson Distribution(4) Geometric Distribution(5) Negative Binomial Distribution(6) Hypergeometric Distribution(7) Multinomial DistributionMath 370X - Lecture 5Frequently Used Discrete DistributionsSaumil PadhyaOctober 11, 2016(1) Uniform DistributionN ≥ 1 is an integerp(x) =1Nfor x = 1, 2, ..., N0 otherwiseE[X] =N + 12V ar[X] =N2− 112(2) Binomial Distributionn ≥ 1 is an integer, and 0 ≤ p ≤ 1p(x) =nx· px· (1 − p)n−xfor x = 0, 1, 2, ..., nE[X] = n · pV ar[X] = n · p · (1 − p)MX(t) = (1 − p + pet)nIn the special case of n = 1, the distribution is referred to as a Bernoulli Distribution.(3) Poisson Distributionλ > 0p(x) =e−λ· λxx!for x = 0, 1, 2, 3, ...E[X] = V ar[X] = λMX(t) = eλ·(et−1)1(4) Geometric Distribution0 ≤ p ≤ 1p(x) = (1 − p)x· p for x = 0, 1, 2, 3, ...E[X] =1 − ppV ar[X] =1 − pp2MX(t) =p1 − (1 − p)etLet Y = X + 1 where X has a geometric distribution.P [Y = y] = P [X = y − 1] = (1 − p)y−1· p for y = 1, 2, 3, ...E[Y ] = E[X] + 1 =1pV ar[Y ] = V ar[X] =1 − pp2(5) Negative Binomial Distributionr > 0 and 0 ≤ p ≤ 1p(x) =r + x − 1x· pr· (1 − p)x=r + x − 1r − 1· pr· (1 − p)xfor x = 0, 1, 2, 3, ...E[X] =r(1 − p)pV ar[X] =r(1 − p)p2MX(t) = p1 − (1 − p)et!rThe special case of r = 1 is the geometric distribution.(6) Hypergeometric Distributionp(x) =Kx·M − Kn − xMnE[X] =nKMV ar[X] =nK(M − K)(M − n)M2(M − 1)2(7) Multinomial Distributionp(x1, x2, x3, .., xk) =n!x1!x2!...xk!· px11px22...pxkkE[Xi] = npiV ar[Xi] = npi(1 − pi)Cov[Xi, Xj] =
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