MATH20812: PRACTICAL STATISTICS I SEMESTER 2 NOTES ON RANDOM VARIABLES Things to Know Rndom Vrible A rndom vrible is function tht ssigns numericl vlue to ech outcome of prticulr experiment. A rndom vrible is denoted by n uppercse letter, such s X nd corresponding lower cse letter such s x is used to denote possible vlue of X. The set of possible numbers of rndom vrible X is referred to s the rnge of X. The probbility of the event tht X = x is denoted by Pr(X = x). Discrete Rndom Vrible A discrete rndom vrible is rndom vrible with finite (or countbly infinite) rnge. Exmples: number of ccidents, number of pplicnt interviewed, number of power plnts, etc Continuous Rndom Vrible If the rnge of rndom vrible contins n intervl of rel numbers, then it is continuous rndom vrible. Exmples: temperture, breking strength, filure time, etc Probbility Mss Function For discrete rndom vrible X: the function f(x) = Pr(X = x) is clled probbility mss function if it stisfies f(x) 0 for ll possible vlues of x nd f(x) = 1. (1) for ll x Probbility Density Function For continuous rndom vrible X: the function f(x) is clled probbility density function if it stisfies f(x) 0 for ll possible vlues of x nd b for ll nd b (see figure 1). Two consequences re: f(x)dx = Pr( < X < b) (2) f(x)dx = Pr( < X < ) = 1 (3) nd f(x)dx = Pr(X = ) = 0. (4) 1
Figure 1 Probbility Density Function of X Pr( < X < b) 0 b X Cumultive Distribution Function (CDF) The cumultive distribution function of rndom vrible X is: F(x) = Pr(X x) = f(y) (5) for ll y x if X is discrete rndom vrible; F(x) = Pr(X x) = if X is continuous rndom vrible. x Properties of CDF The CDF hs the following properties: f(y)dy (6) 2
(i) 0 F(x) 1 (see figure 2); (ii) If b then F() F(b) (see figure 2); (iii) F() = 0 (see figure 2); (iv) F( ) = 1 (see figure 2); (v) If X is continuous rndom vrible then F(b) F() = Pr( < X < b) (see figure 2); (vi) If X is continuous rndom vrible then f(x) = F(x) x. (7) Figure 2 1 Cumultive Distribution Function of X Pr(<X<b) 0 b X Percentiles The 100(1 α)% percentile of rndom vrible X, denoted by x α, is the vlue of X exceeded with probbility α, i.e. Pr(X x α ) = 1 α. (8) 3
Expected Vlue The expected vlue of rndom vrible X is: if X is discrete rndom vrible; E(X) = xf(x) (9) for ll x E(X) = xf(x)dx (10) if X is continuous rndom vrible. Properties of Expecttion (i) E(c) = c (c is constnt); (ii) E(cX) = ce(x) (c is constnt); (iii) E(cX + d) = ce(x) + d (c nd d re constnts). Expecttion of Function For ny rel-vlued function g, the expected vlue of g(x) is: if X is discrete rndom vrible; E(g(X)) = g(x)f(x) (11) for ll x E(g(X)) = g(x)f(x)dx (12) if X is continuous rndom vrible. Vrince The vrince of rndom vrible X is: V r(x) = E [X E(X)] 2 = E(X 2 ) (E(X)) 2. (13) Properties of Vrince (i) V r(c) = 0 (c is constnt); (ii) V r(cx) = c 2 V r(x) (c is constnt); (iii) V r(cx + d) = c 2 V r(x) (c nd d re constnts). 4
Stndrd Devition The stndrd devition of rndom vrible X is: SD(X) = V r(x), (14) mesure of spred. Coefficient of Vrition The coefficient of vrition of rndom vrible X is: CV (X) = SD(X) E(X), (15) dimensionless mesure of spred reltive to the expected vlue. Mesures of Shpe Two dimensionless mesures of shpe re skewness nd kurtosis, defined by γ 1 (X) = E [X E(X)]3 [V r(x)] 3/2 (16) nd γ 2 (X) = E [X E(X)]4 [V r(x)] 2, (17) respectively. Note tht E [X E(X)] 3 ( = E X 3) ( 3E(X)E X 2) + 2 (E(X)) 3 (18) nd E [X E(X)] 4 ( = E X 4) ( 4E(X)E X 3) ( + 6 (E(X)) 2 E X 2) 3 (E(X)) 4. (19) Relibility Function Let rndom vrible X represent the time between filures of system. Clerly this is continuous rndom vrible. The relibility function t time t denoted by F(t) is the probbility tht the system survives longer thn time t, i.e. F(t) = Pr(X > t) = 1 Pr(X t) = 1 F(t). (20) Filure Rte Function The filure rte of mny systems (e.g. humn body) chnge over time. In generl, filure rte is function of the system s lifetime so fr. The hzrd rte or the filure rte function t time t denoted by λ(t) is found by dividing the density function t time t by the relibility function for tht durtion: λ(t) = f(t) F(t). (21) The typicl shpe of hzrd rte function is shown in figure below: Region I, where the function decreses, is termed the region of infnt mortlity; Region II, where the function does not chnge rpidly, is termed the rndom filure re gion; Region III is the wer-out region, where the function increses due to deteriortion. 5
Filure Rte Function 0 5 10 15 20 Region I Region II Region III 0.0 0.2 0.4 0.6 0.8 1.0 t An Alterntive to Kurtosis If X is continuous rndom vrible with pdf f(x) then T f = V r {log (f(x))} (22) mesures the instrinic shpe of the distribution. This mesure ws introduced lst yer (2001) by Dr. K. -S. Song from the Florid Stte University [see the Journl of Sttisticl Plnning nd Inference, volume 93, pp. 51 69]. It is better mesure thn kurtosis in mesuring the shpe of distribution. 6