advertisement

Type Description Finite Random Variable Finite discrete-it can take on only finitely many possible values (ex: X = 0,1,2, or 3). In this case you can list all possible values Probability Distribution Func. Probability mass function (p.m.f.), fX (x) where fX (x) = P(X = x) = height of p.m.f. 0 f X ( x) 1 sum of p.m.f. values = 1 Cumulative Distribution Func. Cumulative density function (c.d.f.), Probability mass function (p.m.f.), fX (x) where fX (x) = P(X = x) In this case, a histogram associated with P( X x) Cumulative density function (c.d.f.), Probability Density function (p.d.f.), The value of the p.d.f., fX (x) Cumulative density function (c.d.f.), FX x P( X x) non-decreasing max value of 1 step function Expected Value, µ Variance, 𝜎 2 Std Dev = 𝜎 E( X ) 𝑉(𝑋) = ∑(𝑥 − 𝜇)2 𝑓𝑋 (𝑥) x P( X x) all x x f X ( x) all x Binomia l Random Variable Continu ous Random Variable Binomial Setting: 1.You have n repeated trials of an experiment. 2. On a single trial, there are two possible outcomes, success or failure. 3.The probability of success, p , is the same from trial to trial. 4.The outcome of each trial is independent. Continuous-if the possible values form an entire interval of numbers (ex: any positive number) Expone ntial Random Variable Exponential random variables are continuous random variables and usually describe the waiting time between consecutive events. Uniform Random Variable If X is uniform on the interval [a,b] then we have a continuous uniform random variable P( X x) . The value of fX (x) is simply the height of the density curve at the value of x. fX (x) > 0 FX x P( X x) E( X ) n p 𝑉(𝑋) = 𝑛𝑝(1 − 𝑝) non-decreasing max value of 1 step function FX ( x) P( X x) P(a X b) FX (b) FX (a) 𝐸(𝑋) = 𝑉(𝑋) = ∫(𝑥 − 𝜇)2 𝑓𝑋 (𝑥)𝑑𝑥 ∞ ∫ 𝑥 ∙ 𝑓𝑋 (𝑥)𝑑𝑥 −∞ P( X x) 1 FX ( x) P( X x) 0 for all x non-decreasing max value of 1 continuous function, usually Probability Density function (p.d.f.), Cumulative density function (c.d.f.), E X 𝑉(𝑋) = 𝛼 2 0 if x 0 0 if x 0 FX ( x) f X ( x) 1 x / if 0 x 1 e x / if 0 x e Probability Density function (p.d.f.), 0 1 f X ( x) b a 0 if x a Cumulative density function (c.d.f.), if x a 0 xa if a x b F X ( x) if a x b b a if x b if x b 1 E( X ) ba 2 𝑉(𝑥) (𝑏 − 𝑎)2 = 12