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Stat 330 Formula Sheet Exam II probability mass function properties: 0 ≤ pX (k) ≤ 1 for all k = 0, 1, 2, . . . and P k pX (k) = 1 probability distribution function properties: FX (t) non-decreasing for t, limt→−∞ FX (t) = 0, limt→∞ FX (t) = 1, step-function, increases at k. expected value & variance properties: E[aX + bY ] = aE[X] + bE[Y ], E[X 2 ] = V ar[X] + (E[X])2 , V ar[aX + b] = a2 V ar[X] V ar[X + Y ] = V ar[X] + V ar[Y ], iff X, Y are independent. E[X · Y ] = E[X] · E[Y ] only if X, Y are independent. P covariance of X and Y : Cov(X, Y ) = (x,y) (x − E[X])(y − E[y])pX,Y (x, y) correlation of X and Y : Corr(X, Y ) = √ Cov(X,Y ) , V ar[X]V ar[Y ] correlation is in [0, 1], unitless. discrete distribution(s): Binomial B(n, p): X = number of successes in n indep. trials, p = P (success) (Bernoulli trials). pmf: pX (k) = nk pk (1 − p)n−k , for FX (k) see Binomial table, E[X] = np, V ar[X] = np(1 − p) Geometric Geo(p): X = number of Bernoulli trials until 1st success, p = P (success). pmf: pX (k) = (1 − p)k−1 p, FX (k) = 1 − (1 − p)k , E[X] = 1/p, V ar[X] = 1−p p2 Poisson P oi(λ): X = number of events in some unit of time/space, λ = rate of events in some unit of time/space. k pmf: pX (k) = e−λ λk! , for FX (k) see Poisson table, E[X] = λ, V ar[X] = λ continuous random variable X uncountable set of possible outcomes (=interval), probability density function : fX (k) = F 0 (X = k), cumulative distribution function FX (t) = P (X ≤ t), R∞ R∞ expected value E[X] = −∞ xfX (x)dx, variance V ar[X] = −∞ (x − E[X])2 fX (x)dx. R∞ probability density function (pdf ) properties: fX (x) ≥ 0 for all x and −∞ fX (x)dx = 1. cumulative distribution function (cdf ) properties: FX (t) non decreasing for t, limt→−∞ FX (t) = 0, limt→∞ FX (t) = 1. rules for densities & distribution functions Rt Rb P (a ≤ X ≤ b) = a fX (x)dx = FX (b) − FX (a), P (X = a) = 0, FX (t) = −∞ fX (x)dx. Uniform U (a, b), X = random value between a and b, all ranges of values in [a, b] are equally likely, pdf: f (x) = 1/(b − a) for x ∈ [a, b], cdf: FX (x) = x−a b−a for x ∈ [a, b], E[X] = (a + b)/2, V ar[X] = (b − a)2 /12. Exponential Exp(λ), X = time/space until 1st occurrence of event, λ = rate of events in some unit of time/space. pdf: f (x) = λe−λx for x ≥ 0, FX (x) = 1 − e−λx for x ≥ 0, E[X] = 1/λ, V ar[X] = 1/λ2 . Exponential distribution is memoryless, i.e. P (X ≤ s + t | X ≥ s) = P (X ≤ t). Erlang Distribution Erlang(k, λ): If Y1 , . . . , Yk are k independent exponential random variables with parameter λ, their sum X has an Erlang distribution: P X := ki=1 Yi is Erlang(k, λ) k is stage parameter, λ is rate parameter Erlang density f (x) = λe−λx · (λx)k−1 (k−1)! for x ≥ 0 E[X] = k/λ, V ar[X] = k/λ2 Erlang cdf is calculated using Poisson cdf: FX (t) = 1 − FY (k − 1) where X ∼ Erlang(k, λ) and Y ∼ P oi(λt) so use Poisson cdf table with parameter= λt Normal r.v.: X ∼ N (µ, σ 2 ), Normal density is “bell-shaped” f (x) = √ 1 e− 2πσ 2 (x−µ)2 2σ 2 E[X] = µ, V ar[X] = σ 2 . standardization: FX (x) = Φ( x−µ σ ); Z ∼ N (0, 1) and Φ(z) ≡ FZ (z) and Φ(−z) = 1 − Φ(z). 2 ) X ∼ N µx , σx2 ), Y ∼ N (µy , σy2 ), then W := aX + bY has normal distribution W ∼ N (µW , σW 2 = a2 σ 2 + b2 σ 2 + 2abCov(X, Y ) where µW = aµx + bµy and σW x y Central Limit Theorem (CLT): If X1 , X2 , . . . , Xn are i.i.d. r.v.’s with E[Xi ] = µ, V ar[Xi ] = σ 2 , P P approx approx then X := n1 ni=1 Xi ∼ N (µ, σ 2 /n) and Sn = i Xi ∼ N (nµ, nσ 2 ) Bin(n, p) P oi(λ) approx ∼ approx ∼ N (np, np(1 − p)) for large n (if np > 5), N (λ, λ) for large λ,