Comps Part I Random Variable

Mar 17, 2024 · 441 words · 1 minute read

总结outline：

1. what is a RV? 🔗

1.1 defn: it’s a function from sample space D to real line.
1.2 randomness: before we do the experiment, we have random experiment, and we assign different prob. to experiment.

2. Discrete RV 🔗

2.1 Bernoulli 🔗

2.2 Binomial: sum of n indep Bernoulli 🔗

sum of two indep binomial (say (m,p) and (n,p)) with same p turns out to be bernoulli with (m+n,p). The PMF can be write out using convolution or MGF.
binomial theorm to proof that probs. sum up to 1, thats why its called binomial dist.

2.3 Hypergeometric 🔗

select c = 5 marbles from A+B = n marbles (A white, B black) without replacement, find the dist. of # of white marbles.
binomial vs. hypergeometric:
- binomial: sample with replacement, trails are indep.
- hypergeometric: sample without replacement, trails are dependent.
- binomial approximation: when n is large (e.g.,100000), c (c = 10) is relatively small, it’s unlikely to select the same marble more than once, then hypergeometric –> binomial. Then we can use binomial approximate hypergeometric

2.4 geometric Geom(p): 🔗

indep. Bern(p) trails, count # of failures before 1st success
sum of geometric series: sum of q^k (k = 0,1,..+infnty)= 1/(1-q), thats why we call it geometric dist.

2.5 negative binomial 🔗

generalization of geometric dist
indep. Bern(p) trails, count # of failures before the r^the success

2.6 poisson (most important discrete dist. in all of stats) 🔗

name after Poisson, a famous French mathematician in 1830s
count the # of ‘success’, where there are a large # of trials, small prob for each trail (e.g., earthquakes in a year in a specific region)
Poisson approximation: Bin(n,p), when n goes to +infnty, p goes to 0, lambda = np is hold constant (lambda is the count, p is the rate parameter, in survival analysis: mu = lambda*Y, where mu is the count, lambda is the rate param, and Y is the total time of follow up.) binomial –> poisson. we can use poisson to approx. binomial.
the rate is a constant

3. continuous dist. 🔗

3.1 uniform 🔗

useful property of Uniform(0,1): F(y) = P(Y <= y) = y
X with CDF F(x), generate X = x = F^-1(U), where U is a uniform(0,1).