Linked Questions

10
votes
3answers
1k views

Inverse transform sampling - CDF is not invertible

Suppose the cumulative distribution function $F$ is given but not invertible to use the inverse transform sampling technique (to compute $X=F^{-1}(Y)$). Do we have other alternative methods? I would ...
8
votes
3answers
411 views

Show that if $X\ge 0$ , $E(X)\le \sum_{n=0}^{\infty}P(X>n)$

If $X$ is a random variable and also let $X\ge 0$. I want to show $E(X)\le \sum_{n=0}^{\infty}P(X>n)$.
7
votes
1answer
6k views

Expectation when cumulative distribution function is given

This is from the book Fundamentals of Probability with Stochastic Processes by Saeed Ghahramani, pages 249-250 which asserts, for any random variable $X$ that is non-negative, expectation of $X$ is ...
2
votes
1answer
3k views

Calculating life time expectancy

How to calculate life time expectancy when not all patients have died. Kaplan-Meier provides a survival curve which is similar to cumulative distribution function but not the actual expectancy. For ...
2
votes
2answers
644 views

What should the integral of a CDF be called?

This is strictly a nomenclature question. I have no particular problem finding double integrals of the type $\int\int\text{pdf}(y) \, d y \,d x$, and I find them quite useful. Whereas we have a good ...
4
votes
2answers
171 views

Test two groups with only sample statistics or without distributional assumptions

I have two sets of samples, A and B. I want to find whether the underlying mean (i.e. if the sample size was infinite) of A is greater than that of B, to a certain confidence (95%). There are two ...
5
votes
1answer
1k views

Conditional Expectation via Integral over Quantile Function

Following this thread "Does a univariate random variable's mean always equal the integral of its quantile function?" I tried to do a similar thing for a conditional expectation. It seems like my ...
5
votes
2answers
134 views

Closed form of the integral of the difference of two Gaussian CDFs?

Problem I'm trying to find the simplest form of the difference of two Guassian CDFs, i.e. $$ \int_{-\infty}^\infty \left( \Phi\left(ax+b \right) - \Phi\left(cx+d \right) \right) dx $$ for $\Phi(...
10
votes
1answer
217 views

References: Tail of the inverse cdf

I am almost sure I have already seen the following result in statistics but I can't remember where. If $X$ is a positive random variable and $\mathbb{E}(X)<\infty$ then $\varepsilon F^{-1}(1-\...
4
votes
1answer
140 views

Sum of products in an expected value

A box contains $n$ balls numbered from 1 to $n$. Suppose you take a ball at a time, putting it back on the box, until you pick a ball twice. How many balls are you expected to take from the box? Let $...
3
votes
1answer
66 views

If all trimmed means are equal does this imply equal distributions?

I am trying to prove the following: Given that $\forall \alpha\in [0,1]$: $$\int_{F_S^{-1}(\alpha)}^{\infty}xf_S(x)\,dx = \int_{F_0^{-1}(\alpha)}^{\infty}yf_0(y)\,dy$$ where $F_S^{-1}(\alpha)$ and $...
1
vote
0answers
58 views

An inequality for the non-negative random variable

I once read the following inequality Is there any specific name for this inequality? And, how to prove it?