Lec_6_Handout\fbox{\parbox{1\textwidth}{When we talk about continuous distributions, we will use a different notation than discrete distributions. The probability density function is f(x) (as opposed to the probability mass function p(x)), distribution function is F(x).}}(1) Uniform Distribution(2) Normal Distribution\fbox{\parbox{1\textwidth}{If X and Y are independent and normally distributed random variables, then all linear combinations of X and Y (for eg. X+Y, X-Y, etc.) are also normally distributed.}}\fbox{\parbox{1\textwidth}{When an exam question asks you for the approximate probability, it is asking you to use the normal approximation.}}(3) Exponential Distribution(4) Gamma DistributionLec 6 FormulasP-05-05tablesMath 370X - Lecture 6Frequently Used Continuous DistributionsSaumil PadhyaOctober 20, 2016When we talk about continuous distributions, we will use a different notation than discretedistributions. The probability density function is f(x) (as opposed to the probability massfunction p(x)), distribution function is F(x).(1) Uniform DistributionAll points in the interval (a, b) have equal probability of occuring. Remember that this is differentfrom thediscreteuniform distribution. If a random variable X has a discrete uniform distribution,X can take values of a countable number of points in the interval (a, b).Note that mean=median for a uniform distribution.(2) Normal DistributionAnormal distribution, X~N(µ, σ2) where X has meanµand varianceσ2. Astandard normaldistributionis a special case of a normal distribution represented by Z~N(0,1) which has a meanof 0 and variance of 1.A normal distribution can be characterized by a symmetric bell-shaped graph. The “flatter” graphhas the larger variance, and is more widely spread around the mean.In the exam, you will be provided with a table of probabilities for astandard normal dis-tribution. So, to calculate probabilities for anormal distributioni.e. for a random variableX~N(µ, σ2), use the following transformation:Z =X − µσSo subtract the mean, divide by the standard deviation, and then use the given table to calculatethe probabilities.If X and Y are independent and normally distributed random variables, then all linear com-binations of X and Y (for eg. X+Y, X-Y, etc.) are also normally distributed.Approximating a distribution using a normal distributionGiven a random variable X with meanµand varianceσ2, probabilities related to distribution ofX can be approximated by assuming that X~N(µ, σ2).1When a normal distribution is used to approximate a discrete distribution,integer correctionshould be applied in the following way: If n and m are integers, the probability P[n≤X≤m] isapproximated using a normal random variable Y with the same mean and variance as X, andthen finding the probability P[n -12≤Y≤m +12]. In other words, we extend the interval [n, m]to [n -12,m +12].When an exam question asks you for the approximate probability, it is asking you to use thenormal approximation.(3) Exponential DistributionThe exponential distribution is often used as a model for the time until some specific event occurs,say the time until the next earthquake at a certain location.Lack of Memory: P[X > x+y | X > x] = P[X > y]Exponential and Poisson: Consider an event for eg. the arrival of a new insurance claim at aninsurance office. Let X represent the time between arrival of a claim. Let N represent the numberof claims that have occured after one unit of time. If X~Exp(mean=1/λ), then N~Poi(mean=λ).(4) Gamma DistributionThe gamma function is defined as:Γ(α) =∞Z0yα· e−ydyΓ(n) = (n − 1)! if n is a positive integerΓ(1/2) =√πNote that the exponential distribution is a special case of a gamma distribution withα= 1 andβ= λ2Scanned by
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