Review of Prob. & Stat. Infer.
Chap. 2 Discrete Dist.
1. The moment-generating function (m.g.f): M(𝑡) = 𝐸(𝑒 𝑡𝑋 ).
2. The negative binomial dist.: r success in a series of Bernoulli trials.
𝑥 − 1 𝑟 𝑥−𝑟
g(𝑥) = (
)𝑝 𝑞
𝑟−1
When computing the m.g.f, use the Maclaurin’s series expansion of the
𝑥 − 1 𝑥−𝑟
function: h(𝑤) = (1 − 𝑤)−𝑟 = ∑∞
) 𝑤 , where x = r + n.
𝑥=𝑟 (
𝑟−1
3. The Poisson dist.: x changes occur in a unit interval.
An approximate Poisson process: Let the nu. of changes that occur in
a given continuous interval be counted. Then we have an approximate
Poisson process with parameter λ > 0 if the following conditions are
satisfied:
(1) The nu. of changes occurring in nonoverlapping intervals are
independent.
(2) The prob. of exactly one change occurring in a sufficiently short
interval of length h is approximately λh.
(3) The prob. of two or more changes occurring in a sufficiently short
interval is essentially zero.
According to the assumptions described above, we have Poisson dist.
as 𝑓(𝑥) =
λ𝑥 𝑒 −λ
𝑥!
, 𝑥 = 0, 1, 2 … , where λ > 0.
𝐸(𝑋) = 𝑉𝑎𝑟(𝑋) = 𝜆.
The number of occurrences of X in the interval of length of t has the
Poisson p.m.f.
(λt)𝑥 𝑒 −λt
𝑓(𝑥) =
, 𝑥 = 0, 1, 2, … .
𝑥!
Here we treat the interval of length t as if it were the “unit interval”
with mean λt instead of t.
We can use the Poisson dist. to approximate the Binomial dist. with
large 𝑛 and small 𝑝, where λ = 𝑛𝑝.
Chap 3. Continuous Dist.
1. Quantile, quartile, percentile:
A group of data runs from y0 to yn+1, and y1, y2, …, yn divide [y0, yn+1] into
(n + 1) intervals of equal length, then yr is called
the quantile of order r/(n + 1),
the 100r/(n + 1)th percentile.
The first quartile is the quantile of 1/4, or the 25th percentile, the
second quartile (median) …, the third quartile ….
Another explanation of percentile: The (100p)th percentile is a number
πp such that the area under f(x) to the left ofπp is p. That is
𝜋𝑝
𝑝 = ∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝜋𝑝 ).
−∞
q-q plot: Assume the theoretical distribution is a good model for the
observations, then 𝑦𝑟 ≈ 𝜋𝑝 , where 𝑝 = 𝑟/(𝑛 + 1). Then (𝑦𝑟 , 𝜋𝑝 )
is on the line y = x. We can use this method to check if the given data
satisfies a certain dist.
2. The uniform dist. (omitted)
3. The exponential dist.
Let W be the waiting time in a Poisson process with parameter λ > 0,
and explore the dist. of W.
𝐹(𝑤) = 𝑃(𝑊 ≤ 𝑤) = 1 − 𝑃(𝑊 > 𝑤)
= 1 − 𝑃(𝑛𝑜 𝑐ℎ𝑎𝑛𝑔𝑒𝑠 𝑖𝑛 [0, 𝑤])
We treat [0, w] as a unit interval, then we have the p.m.f
(λ𝑤)𝑥 𝑒 −λ𝑤
𝑔(𝑥) =
𝑥!
𝑃(𝑛𝑜 𝑐ℎ𝑎𝑛𝑔𝑒𝑠 𝑖𝑛 [0, 𝑤]) = 𝑔(0) = 𝑒 −λ𝑤
Hence, the c.d.f. of W is
𝐹(𝑤) = 1 − 𝑒 −λ𝑤
the p.d.f. of W is
𝑓(𝑤) = 𝐹 ′ (𝑤) = λ𝑒 −λ𝑤 =
1 −𝑥⁄θ
𝑒
,0 ≤ 𝑥 < ∞
θ