Key discrete-type distributionsBernoulli(p): 0 ≤ p ≤ 1pmf: p(i) =p i = 11 − p i = 0mean: p variance: p(1 − p)Example: One if heads shows and zero if tails shows for the flip of a coin. The coin is called fair ifp =12and biased otherwise.Binomial(n, p): n ≥ 1, 0 ≤ p ≤ 1pmf: p(i) =nipi(1 − p)n−i0 ≤ i ≤ nmean: np variance: np(1 − p)Significance: Sum of n independent Bernoulli random variables with parameter p.Poisson(λ): λ ≥ 0pmf: p(i) =λie−λi!i ≥ 0mean: λ variance: λExample: Number of phone calls placed during a ten second interval in a large city.Significant property: The Poisson pmf is the limit of the binomial pmf as n → +∞ and p → 0 insuch a way that np → λ.Geometric (p) : 0 < p ≤ 1pmf: p(i) = (1 − p)i−1p i ≥ 1mean:1pvariance:1 − pp2Example: Number of independent tosses of a coin until heads first appears.Significant property: If L has the geometric distribution with parameter p, P {L > i} = (1 −p)iforintegers i ≥ 1. So L has the memoryless property in discrete time:P {L > i + j | L > i} = P {L > j} for i, j ≥ 0.Any positive integer-valued random variable with this property has the geometric distribution forsome p.Key continuous-type distributionsGaussian or Normal(µ, σ2) µ ∈ R, σ ≥ 0pdf : f (u) =1√2πσ2exp−(u −µ)22σ2mean: µ variance: σ2Notation: Q(c) = 1 −Φ(c) =R∞c1√2πe−u22duSignificant property (CLT): For independent, indentically distributed r.v.’s with mean mean µ, variance σ2:limn→∞PX1+ ··· + Xn− nµ√nσ2≤ c= Φ(c)Exponential(λ)pdf: f(t) = λe−λtt ≥ 0 mean:1λvariance:1λ2Example: Time elapsed between noon sharp and the first time a telephone call is placed after that, in a city,on a given day.Significant property: If T has the exponential distribution with parameter λ, P {T ≥ t} = e−λtfor t ≥ 0. SoT has the memoryless property in continuous time:P {T ≥ s + t | T ≥ s} = P {T ≥ t} s, t ≥ 0Any nonnegative random variable with the memoryless property in continuous time is exponentially dis-tributed.Uniform(a, b): −∞ < a < b < ∞pdf: f(u) =1b−aa ≤ u ≤ b0 elsemean:a + b2variance:(b − a)212Erlang(r, λ): r ≥ 1, λ ≥ 0pdf: f(t) =λrtr−1e−λt(r − 1)!t ≥ 0 mean:rλvariance:rλ2Significant property: The distribution of the sum of r independent random variables, each having theexponential distribution with parameter λ. (If r > 0 is real valued and (r − 1)! is replaced by Γ(r) thegamma distribution is obtained.)Rayleigh(σ2): σ2> 0pdf: f(r) =rσ2exp−r22σ2r > 0 CDF : 1 − exp−r22σ2mean: σrπ2variance: σ22 −π2Example: Instantaneous value of the envelope of a mean zero, narrow band noise signal.Significant property: If X and Y are independent, N(0, σ2) random variables, then (X2+ Y2)12has theRayleigh(σ2) distribution. Failure rate function is linear: h(t)