Recall also from Calculus, that areas under the curves areapproximated by Riemann sums..(xi,xi+1](xi,xi+1]P(xi<X≤xj)(xi,xi+1](xj−1,xj]P(xi<X≤xj)(xi,xi+1](xj−1,xj]A≃n∑i=1f(x*i)ΔxiAn “ideal” histogram of X= # of heads among 10 coins

Whena Riemann sum becomes a definte integral,. Recall from Calculus,that, whereis the indefinite integral of ,,meaning, and that.Recall that every discrete random variable X has a probability mass function.For any set A,.Any continuous random variable X has aprobability density functionf(x) (which is theequivalent of the probability mass function) and such that for any,Hence, as in calculus, a sum becomes an integral. The time came to refresh your memoryabout all those derivatives and integration you once learned in your calculus class!For example, a random X from an interval (a,b) is called theuniform random variableover(a,b). (We saw it in the past in problems about a bus arriving at random in a certain timeinterval). It turns out its probability density function isThe probability density of the uniform random variable over (1,3):Then, according to the above formula,is the area under f betweenand ,which agrees with the formulawe used in the past.To recap, X is a continuous random variable, if there is f(x) such that equationholds for any.In particular, forwe get.Consequently, for any continuous random variable X,for every .Δx→0∫baf(x)dx∫βαf(x)dx= [F(x)]βαF(x)f F(x) =∫f(x)dxF′(x) =f(x)[F(x)]βα=F(β)−F(α)p(x) =P(X=x)P(X∈A) =∑a∈Ap(a)α≤β( * )P(α<X≤β) =∫βαf(x)dx.f(x) ={1b−aforx∈[a,b]0otherwise.P(α<X≤b)αβP(α<X≤β) =∫βα1b−adx=[xb−a]βα=βb−a−αb−aP(α<X≤β) =β−αb−a( * )α,βα=βP(X=α) =P(α<X≤α) =∫ααf(x)dx=F(α)−F(α) = 0P(X=α) = 0α

That is the opposite behavior of the discrete random variables! Not every random variable isdiscrete or continuous. In Sec. 4.1 of these notes we discussed an example of a randomvariable of a mixed type, which is neither of these two types. But for the remainder of thiscourse we will focus on the continuous random variable.Hence, for continuous random variablesWe mentioned thatcan be obtained from a histogram when the bin size. Anotherway to understandmore intuitively: note that for any and any small ,.Hence, f(x) is the probability of X lying in a small interval containing x,Divided by the length of that interval.Some properties of:Lemmafor every .Proof: Ifthen, assuming f is continuous,for every x in some smallneighborhoodofand then,which is impossible.We also have. In other words the total area underis 1.Conversely, one can prove that any functionwith the total area underone, corresponds to somecontinuous random variable X.For example, this is a probability density function for some X:P(X<2)=area under the graph betweenand 2= areabetween 1 and 2 =An analogous statement for discrete random variables and the probability mass was discussedin Sec 4.3 of these notes.

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Term

Fall

Professor

Kuhlmann

Tags

Probability theory