重要的离散型随机变量

待整理：

1. 离散型随机变量-伯努利随机变量
   1.1 伯努利分布
   1.2 几何分布
   1.3 二项分布
   1.4 泊松分布
2. 连续性随机变量
   2.1 均匀随机变量
   2.2. 指数随机变量
   2.3. 指数随机变量无记忆性
   2.4. 中心极限定理，引出正态分布和高斯分布
3. 随机变量函数

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Bernouillirandom variables are used to model experiments that have two possible outcomes.By convention we usually represent an outcome by 0 and the other outcome by 1. A canonicalexample is flipping a biased coin, such that the probability of obtaining heads isp. If weencode heads as 1 and tails as 0, then the result of the coin flip corresponds to a Bernouillirandom variable with parameterp.

Definition 5.4(Bernouilli).The pmf of a Bernouilli random variable with parameterp∈[0,1]is given by
pX(0) = 1−p,(53)
pX(1) =p.(54)

Definition 5.6(Geometric).The pmf of a geometric random variable with parameterpisgiven by
pX(k) = (1−p)k−1p, k= 1,2,...(56)

Definition 5.7(Binomial).The pmf of a binomial random variable with parametersnandpis given by
pX(k) =(nk)pk(1−p)(n−k), k= 0,1,2,...,n.(57)

Definition 5.11(Poisson).The pmf of a Poisson random variable with parameterλis givenby
pX(k) =λke−λk!, k= 0,1,2,..

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A uniformrandom variable models an experiment in which every outcome within a contin-uous interval is equally likely
Definition 5.17(Uniform).The pdf of a uniform random variable with domain[a,b]isgiven by
fX(x) ={1/ b−a,ifa≤x≤b,
0,otherwise.

The exponential random variable is often used to modelwaiting times: the time it takesuntil a certain event occurs. Examples of such events include the decay of a radioactiveparticle, a telephone call or the mechanical failure of a device.Definition 5.18(Exponential).The pdf of an exponential random variable with parameterλis given byfX(x) ={λe−λx,ifx≥0,0,otherwise.(78)An important property of an exponential random variable is that it ismemoryless. Intu-itively this means that knowing the time you have waited up to now gives younoinformationabout how much more time you will have to wait (a feeling you might have felt calling cus-tomer service)

Lemma 5.19(Exponential random variables are memoryless).LetTbe an exponentialrandom variable, for anyt > t0P (T≤t−t0) = P (T≤t|T > t0),(79)i.e. the waiting time starting at anyt0is distributed exactly like the original waiting time.

The Gaussian or normal random variable is arguably the most notorious in probabilityand statistics. It is often used to model variables with unknown distributions in the naturalsciences. This is motivated by the fact that sums of independent random variables convergeto Gaussian distributions under certain assumptions. This phenomenon is captured by theCentral Limit Theorem, which we will discuss further on in the course

5.3 Functions of random variablesWithin a probabilistic model an uncertain quantityYmay be well modeled as a function ofanother quantityXwith known distribution:Y=g(X). IfXis discrete, by definition ofthe pmf,pY(y) = P (Y=y)(86)= P (g(X) =y)(87)=∑{x|g(x)=y}pX(x).(88)IfXis continuous, thenFY(y) = P (Y≤y)(89)= P (g(X)≤y)(90)=∫{x|g(x)≤y}fX(x) dx,(91)19

where the last equality only holds if the cdf ofXis differentiable.

参考：https://cims.nyu.edu/~cfgranda/pages/DSGA1002_fall15/material/probability_1.pdf