1 Probability Density Functions


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1 Lis Yn CS 9 Continuous Distributions Lecture Notes #9 July 6, 28 Bsed on chpter by Chris Piech So fr, ll rndom vribles we hve seen hve been discrete. In ll the cses we hve seen in CS 9, this ment tht our RVs could only tke on integer vlues. Now it s time for continuous rndom vribles, which cn tke on vlues in the rel number domin (R). Continuous rndom vribles cn be used to represent mesurements with rbitrry precision (e.g., height, weight, or time). Probbility Density Functions In the world of discrete rndom vribles, the most importnt property of rndom vrible ws its probbility mss function (PMF), which told you the probbility of the rndom vrible tking on certin vlue. When we move to the world of continuous rndom vribles, we re going to need to rethink this bsic concept. If I were to sk you wht the probbility is of child being born with weight of exctly kilogrms, you might recognize tht question s ridiculous. No child will hve precisely tht weight. Rel vlues re defined with infinite precision; s result, the probbility tht rndom vrible tkes on specific vlue is not very meningful when the rndom vrible is continuous. The PMF doesn t pply. We need nother ide. In the continuous world, every rndom vrible hs probbility density function (PDF), which sys how likely it is tht rndom vrible tkes on prticulr vlue, reltive to other vlues tht it could tke on. The PDF hs the nice property tht you cn integrte over it to find the probbility tht the rndom vrible tkes on vlues within rnge (, b). X is continuous rndom vrible if there is function f (x) for x, clled the probbility density function (PDF), such tht: P( X b) = b dx f (x) To preserve the xioms tht gurntee P( X b) is probbility, the following properties must lso hold: P( X b) P( < X < ) =
2 2 A common misconception is to think of f (x) s probbility. It is insted wht we cll probbility density. It represents probbility divided by the units of X. Generlly this is only meningful when we either tke n integrl over the PDF or we compre probbility densities. As we mentioned when motivting probbility densities, the probbility tht continuous rndom vrible tkes on specific vlue (to infinite precision) is. P(X = ) = dx f (x) = This is very different from the discrete setting, in which we often tlked bout the probbility of rndom vrible tking on prticulr vlue exctly. 2 Cumultive Distribution Function Hving probbility density is gret, but it mens we re going to hve to solve n integrl every single time we wnt to clculte probbility. To sve ourselves some effort, for most of these vribles we will lso compute cumultive distribution function (CDF). The CDF is function which tkes in number nd returns the probbility tht rndom vrible tkes on vlue less thn (or equl to) tht number. If we hve CDF for rndom vrible, we don t need to integrte to nswer probbility questions! For continuous rndom vrible X, the cumultive distribution function is: F X () = P(X ) = dx f (x) This cn be written F(), without the subscript, when it is obvious which rndom vrible we re using. Why is the CDF the probbility tht rndom vrible tkes on vlue less thn (or equl to) the input vlue s opposed to greter thn? It is mtter of convention. But it is useful convention. Most probbility questions cn be solved simply by knowing the CDF (nd tking dvntge of the fct tht the integrl over the rnge to is ). Here re few exmples of how you cn nswer probbility questions by just using CDF: Probbility Query Solution Explntion P(X ) F() This is the definition of the CDF P(X < ) F() Note tht P(X = ) = P(X > ) F() P(X ) + P(X > ) = P( < X < b) F(b) F() F() + P( < X < b) = F(b)
3 3 As we mentioned briefly erlier, the cumultive distribution function cn lso be defined for discrete rndom vribles, but there is less utility to CDF in the discrete world, becuse with the exception of the geometric rndom vrible, none of our discrete rndom vribles hd closed form (tht is, without ny summtions) functions for the CDF: F X () = P(X = i) Exmple Let X be continuous rndom vrible (CRV) with PDF: i= C(4x 2x 2 ) when < x < 2 otherwise In this function, C is constnt. Wht vlue is C? Since we know tht the PDF must sum to : 2 dx C(4x 2x 2 ) = ) C (2x 2 2x3 2 = 3 x= (( C 8 6 ) ) = 3 Solving this eqution for C gives C = 3/8. Wht is P(X > )? dx f (x) = 2 dx 3 8 (4x 2x2 ) = 3 8 (2x 2 2x3 3 ) 2 x= = 3 8 [( 8 6 ) ( 2 2 )] = Exmple 2 Let X be RV representing the number of dys of use before your disk crshes, with PDF: First, determine λ. Recll tht Ae Au du = e Au : Wht is P(X < )? F() = λ λe x/ when x otherwise dx λe x/ = dx e x/ = λ e x/ x= = λ = λ = / dx e x/ = e x/ x= = e / +.95
4 4 3 Expecttion nd Vrince For continuous RV X: E[X] = E[g(X)] = E[X n ] = For both continuous nd discrete RVs: dx x f (x) dx g(x) f (x) dx x n f (x) E[X + b] = E[X] + b Vr(X) = E[(X µ) 2 ] = E[X 2 ] (E[X]) 2 Vr(X + b) = 2 Vr(X) (with µ = E[X]) 4 Uniform Rndom Vrible The most bsic of ll the continuous rndom vribles is the uniform rndom vrible, which is eqully likely to tke on ny vlue in its rnge (α, β). X is uniform rndom vrible (X Uni(α, β)) if it hs PDF: f (x) = β α when α x β otherwise Notice how the density /( β α) is exctly the sme regrdless of the vlue for x. Tht mkes the density uniform. So why is the PDF /( β α) nd not? Tht is the constnt tht mkes it such tht the integrl over ll possible inputs evlutes to. The key properties of this RV re: b P( X b) = dx f (x) = b (for α b β) β α β x E[X] = dx x f (x) = dx β α = x 2 β 2( β α) = α + β 2 Vr(X) = ( β α)2 2 α x=α
5 5 5 Exponentil Rndom Vrible An exponentil rndom vrible (X Exp(λ)) represents the time until n event occurs. It is prmetrized by λ >, the (constnt) rte t which the event occurs. This is the sme λ s in the Poisson distribution; Poisson vrible counts the number of events tht occur in fixed intervl, while n exponentil vrible mesures the mount of time until the next event occurs. (Exmple 2 snekily introduced you to the exponentil distribution lredy; now we get to use formuls we ve lredy computed to work with it without integrting nything.) Properties The probbility density function (PDF) for n exponentil rndom vrible is: λe λx if x else The expecttion is E[X] = λ nd the vrince is Vr(X) = λ 2 There is closed form for the cumultive distribution function (CDF): F(x) = e λx where x Exmple 3 Let X be rndom vrible tht represents the number of minutes until visitor leves your website. You hve clculted tht on verge visitor leves your site fter 5 minutes, nd you decide tht n exponentil distribution is pproprite to model how long person stys before leving the site. Wht is the P(X > )? We cn compute λ = 5 either using the definition of E[X] or by thinking of how mny people leve every minute (nswer: onefifth of person ). Thus X Exp(/5). P(X > ) = F() = ( e λ ) = e Exmple 4 Let X be the number of hours of use until your lptop dies. On verge lptops die fter 5 hours of use. If you use your lptop for 73 hours during your undergrdute creer (ssuming usge = 5 hours/dy nd four yers of university), wht is the probbility tht your lptop lsts ll four yers? As bove, we cn find λ either using E[X] or thinking bout lptop deths per hour: X Exp( 5 ). P(X > 73) = F(73) = ( e 73/5 ) = e
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