4.1. Probability Density Functions


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1 STT Proility Density Functions Ojectives. Continuous rndom vrile  vers  discrete rndom vrile. Proility density function. Uniform distriution nd its properties. Expected vlue nd vrince of continuous rndom vriles. Description of continuous rndom vriles (crv) A continuous rndom vrile tkes on n uncountly infinite numer of vlues. The set of possile vlues of every crv is n entire intervl, finite or infinite. Continuous rndom vriles usully re mesurements. Exmples of continuous rndom vriles Weight nd height of people. Depth mesure in certin loction of lke. The mount of sugr in n ornge. The time required to run mile. The computer time (in seconds) required to process certin progrm. Mesurement of ny physicl quntity with dditive rndom error (noise). The mount of rin flls in the certin city over fixed period of time. Definition Let X e continuous rndom vrile. The proility density function of X is function f such tht for ny two numers nd with P X f x dx 1
2 STT Geometric interprettion of P X f x dx, P X Are under the curve y f x ove the intervl Properties of Proility Density Function. f x f xdx 1 for ll x Exmple (#1 p.14 textook) The current in certin circuit s mesured y n mmeter is continuous rndom vrile X with the following density function: f x.7x x otherwise ) Verify tht the totl re under the density curve is indeed 1. ) Clculte P X 4. 4 P X 4.7x dx.6 c) Clculte P. X P. X 4..7x dx. d) Clculte P X P X 4..7x dx
3 STT Continuous rndom vrile X tkes every its vlue with proility. P X x ues x for ll possile vl ecuse c c P X c f x dx lim f x dx c c Exmple (continution of 4.1.6) P X 4 nd 4 P. X 4. P X 4. nd P X 4.. P X ; nd P. X 4. In ech pir ove, the numers re equl Uniform Distriution. ; Definition. A continuous rv X is sid to hve uniform distriution on the intervl, if the pdf of X is 1 x f x;, otherwise Exmple Assume the witing time t us stop is uniformly distriuted on the intervl [, ]. Find the proility tht it is etween 1 nd minutes. Solution. 1 x f x;, otherwise 1 1 P1 X dx 1 1
4 STT Exmple Let X e the time hedwy for two rndomly chosen consecutive crs on freewy during period of hevy flow. The pdf of X cn e pproximted y f x.1 x..1e x. otherwise Wht is the proility tht hedwy time is t most sec? Solution. P X f x dx..1 e.1e Exmple x..1t.1t e dt 1 e dx A college professor never finishes his lecture efore the end of the hour nd lwys finishes his lectures within min fter the hour. Let the time tht elpses etween the end of the hour nd the end of the lecture nd suppose the pdf of X is kx x f x otherwise ) Find the vlue of k nd drw the corresponding density curve. Hint: Totl re under the grph of f x is 1. ) Wht is the proility tht the lecture ends within 1 min of the end of the hour? c) Wht is the proility tht the lecture continues eyond the hour for etween 6 nd 9 sec? d) Wht is the proility tht the lecture continues for t lest 9 sec eyond the end of the hour? 4
5 STT Solution. k ) kx dx 1 x 1 k 8 1 ) P X 1 x dx c) P1 X 1. x dx d) P X Numericl Chrcteristics of Continuous RVs Formuls of E X nd E h X for Continuous RVs. If X is continuous rndom vrile with pdf f f x nd h h X, then X E X x f x dx hx E h X h x f x dx A mode of continuous proility distriution is often considered to e ny vlue x t which its proility density function hs loclly mximum vlue, so ny pek is mode.
6 STT Formuls of V X nd X for continuous rv. Shortcut formul X V X X V X x f x dx ; V X E X E X E X x f x dx The vrince nd stndrd devition give quntittive mesures of how much spred there is in the distriution or popultion of x vlues. Agin is roughly the size of typicl devition from. Shortcut formul for in the cse of continuous rndom vriles is the sme s in discrete cse Computtionl exmples. 1) Uniform distriution on,. 1 x f x;, otherwise E X 1 1 x 1 xdx 1 1 x 1 E X x dx V X E X E X SD
7 STT ) Given f x cx x 1, otherwise Find the vlue of constnt c, then compute E X, V X, nd X. 1 1 x c cxdx 1 c 1 c E X E X 1 xdx 1 x dx V X E X E X 9 9 ) Given f x cos x x, /. otherwise Find E X, V X, ; EX, V X / E X xcos xdx V X E X E X / 8 E X x cos xdx 4 7
8 STT Cumultive Distriution Function of Continuous RV Ojectives. Definition of cumultive distriution function for continuous rv. Otining density from cumultive distriution function. Computing proilities y using density nd cumultive distriution functions Definition. The cumultive distriution function F for continuous rndom vrile X is defined for every numer x y x F x P X x f tdt For ech x, F(x) is the re under the density curve to the left of x. p. 144 textook 4... Connection etween f from F. If X is continuous rv with pdf f nd cdf F, then t every x t which the derivtive F exists, F x f x 8
9 STT Exmples for computing. #1. Uniform distriution on., Given pdf f for uniform distriution on., Find corresponding cdf F. 1 f x;, Solution. X otherwise F?, x 1 F dx, x ny rel numer, F, x x 1 x dt x 1 x #. Given pdf f for continuous rv X. f x x 1 x1 otherwise Find corresponding cdf F. F, x x 1 1 t dt x 1 1 x 1 1 x 1 9
10 STT Percentiles of Continuous Distriution. Let p e numer etween nd 1. The (1p)th percentile of the distriution of continuous rv X, denoted y p, is defined y p p F p f t dt In prticulr, the medin of continuous distriution, denoted, is the th percentile, so stisfies the eqution. F. Geometric mening of lst eqution. Hlf the re under density curve is to the left of nd hlf is to the right of Exmples of density function with symmetry out some point. The medin nd the men (if it exists) of symmetric distriution oth occur t the point out which the symmetry occurs. If symmetric distriution is unimodl, the mode coincides with the medin nd men Norml Rndom Vrile Exmple of Symmetric Distriution. A continuous rv X is sid to hve norml distriution with prmeters nd (or nd ), where nd, if the pdf of X is f x x 1 e x For this distriution, the medin, mode, nd expected vlue re the sme. 1
11 STT
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