Linked Questions
13 questions linked to/from Does a univariate random variable's mean always equal the integral of its quantile function?
14
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3
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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 ...
- 307
9
votes
1
answer
13k
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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
...
- 193
8
votes
3
answers
665
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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 $E(X)\le \sum_{n=0}^{\infty}P(X>n)$.
- 403
4
votes
2
answers
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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 ...
- 13.3k
2
votes
1
answer
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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 ...
- 123
5
votes
2
answers
364
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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(...
- 1,565
4
votes
2
answers
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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 ...
- 7,139
5
votes
1
answer
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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 ...
- 1,082
0
votes
2
answers
1k
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Calculating Expected Value from CDF
I have two questions if someone can help me and give me reference used.
Can we calculate Expected Value (EV) by reading random variables from Cumulative Distribution Function (CDF)? For example, P90 = ...
10
votes
1
answer
281
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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-\...
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4
votes
1
answer
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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 $...
- 425
1
vote
0
answers
166
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An equality for expectation of the non-negative random variable [duplicate]
I once read the following inequality
Is there any specific name for this inequality? And, how to prove it?
- 3,089
3
votes
1
answer
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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 $...
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