关于数学统计方面pdf和cdf的问题（题目是英文的）

Question

1. The random variable, X has the following probability mass function.
p(x) = c/x(x + 1) (x = 1, 2, 3, . . .)
(i) Find the value of the constant c.
(ii) Find the cumulative distribution function of X and sketch both the
probability mass function and the cumulative distribution function.
NOTE: Think carefully about the values of x for which you need to define
the distribution function.
(iii) Calculate P(X ≥ 50) and P(X ≥ 50|X ≥ 40).

2.A continuous random variable X has the following probability density function
fX(x) =0, for x < 0
1/3, for 0 ≤ x < α
4/3, for α ≤ x < 2α
1/3, for 2α ≤ x < 3α
0, x ≥ 3α.
(a) Find the value of the constant α.
(b) Write down the Cumulative Distribution Function (CDF) of X.
(c) Sketch a rough plot, indicating points of intersection, of the probability
density function fX(x).
(d) Hence, or otherwise, calculate E(X).

Answer 1

这个是你的作业吧。我给你个提示，就不给你说答案了。

cdf是cumulative distribution function（F(x)）,所以可以理解为是在一定区间里所有pdf(f(x))相加之后的结果。所以他们相加的结果是1。所以直接integrate p(x)，就是cdf。F(x) = integral(f(x))=integral(p(x))= 1。然后c就求出来了。作图就比较简单了，一定记住cdf是cumulative，所以是每个p(x)相加，就是y轴的值。第三问就是P(X ≥ 50) is the integration of pdf from 50 to infinity ,你也可以用1-p(x<50)来算。另外一个算出p(X ≥ 40)，然后相除就可以了。

第二问我帮你把cdf的图画出来了，alpha也算出来了，至于cdf的式子还有其他的还是你自己算吧，图已经出来了，其实很简单了。