54
votes
Accepted
Why is the sum of two random variables a convolution?
Convolution calculations associated with distributions of random variables are all mathematical manifestations of the Law of Total Probability.
In the language of my post at What is meant by a “...
- 334k
52
votes
Accepted
Intuitive explanation of Kolmogorov Smirnov Test
The Kolmogorov-Smirnov test assesses the hypothesis that a random sample (of numerical data) came from a continuous distribution that was completely specified without referring to the data.
Here is ...
- 334k
50
votes
Why is the sum of two random variables a convolution?
Notation, upper and lower case
https://en.wikipedia.org/wiki/Notation_in_probability_and_statistics
Random variables are usually written in upper case roman letters: $X$, $Y$, etc.
Particular ...
- 86.5k
46
votes
Why is the sum of two random variables a convolution?
Your confusion seems to arise from conflating random variables with their distributions.
To "unlearn" this confusion, it might help to take a couple of steps back, empty your mind for a moment, ...
- 2,084
34
votes
Accepted
Empirical CDF vs CDF
Let $X$ be a random variable.
The cumulative distribution function $F(x)$ gives the $P(X \leq x)$.
An empirical cumulative distribution function function $G(x)$ gives $P(X \leq x)$ based on the ...
- 23k
27
votes
Accepted
Is the CDF of the Mean always 0.5 for all kind of distributions?
No, this is false. That point is the median, and it is not equal to the mean in all cases, for example, an exponential distribution.
$$
X\sim \exp(1)\\
\mathbb E[X]=1\\
\operatorname{median}(X)=\log 2\...
- 67.1k
23
votes
How to measure the shift between two cumulative distribution functions (CDFs)?
The absolute value of this area is
$$\int_{x=-\infty}^\infty \lvert F(x) - G(x)\rvert \,\mathrm{d}x,$$
which note – at least for continuous distributions – is exactly equal to
$$\int_{x=-\infty}^\...
- 25.2k
23
votes
Accepted
Distribution of maximum of normally distributed random variables
Problems like this, where you want to differentiate the product of a bunch of functions that depend on your variable of interest, can be dealt with by logarithmic differentiation. Let $\Phi$ and $\...
- 133k
22
votes
Inverse transform sampling - CDF is not invertible
The inverse cdf method operates even when the cdf is not invertible, using the generalised inverse$$F^-(u)=\sup\{x;\ F(x)\le u\}\tag{1}$$
which is always defined for $u\in(0,1)$.
When there is no ...
- 108k
22
votes
Accepted
PDF does not integrate to 1 - where is my mistake?
As pointed out in comments, the range of integration in your integral does not match the listed support of the random variable (which is $\mu \leqslant x < \infty$). Start by correcting the ...
- 133k
19
votes
Find CDF from an estimated PDF (estimated by KDE)
There's no need to integrate anything if you know the cdf of the kernel itself. I believe this is straightforward for all the common kernels.
Note that
a KDE is a mixture density
the cdf of a ...
- 290k
17
votes
Empirical CDF vs CDF
Is there any difference between Empirical CDF and CDF?
Yes, they're different. An empirical cdf is a proper cdf, but empirical cdfs will always be discrete even when not drawn from a discrete ...
- 290k
17
votes
Why is the Empirical Distribution based on the Cumulative Distribution?
The empirical distribution function $\hat{F}(\cdot)$
is a step function by construction, since it puts a probability (Dirac) mass of $1/n$ on every term in the sample, $(x_1,\ldots,x_n)$, hence jumps ...
- 108k
16
votes
Accepted
Cdf of minimum of two iid random variables
It's a common trick. If $X = \min(Y_1,Y_2)$ and $F$, $F_X$ are the CDFs of the $Y_i$s and $X$, respectively, then
\begin{align*}
F_X(x) &= 1 - P(X > x) \\
&= 1- P(Y_1 > x, Y_2 > x) \\...
- 21.5k
16
votes
Accepted
Rationale behind defining distribution function with strict inequality
In Section "1.1 Monotone Functions" of Chung's classic probability textbook A Course in Probability Theory, it is stated:
How can $f_1$ and $f_2$ differ at all? This can happen only when $...
- 21.2k
15
votes
Empirical CDF vs CDF
The empirical CDF is built from an actual data set (in the plot below, I used 100 samples from a standard normal distribution). The CDF is a theoretical construct - it is what you would see if you ...
- 3,792
15
votes
What is the intuitive meaning behind plugging a random variable into its own pdf or cdf?
A transform of a random variable $X$ by a measurable function $T:\mathcal{X}\longrightarrow\mathcal{Y}$ is another random variable $Y=T(X)$ which distribution is given by the inverse probability ...
- 108k
15
votes
Is the CDF of the Mean always 0.5 for all kind of distributions?
The mean is the point where the area left and right between the CDF and the vertical line through the mean are equal.
For symmetric distributions, the left and the right side/area have the same shape ...
- 86.5k
14
votes
Why is the CDF of a sample uniformly distributed
Here's some intuition. Let's use a discrete example.
Say after an exam the students' scores are $X = [10, 50, 60, 90]$. But you want the scores to be more even or uniform. $h(X) = [25, 50, 75, 100]$ ...
- 141
14
votes
Distribution function terminology (PDF, CDF, PMF, etc.)
As noted by Wikipedia, probability distribution function is ambiguous term:
A probability distribution function is some function that may be used
to define a particular probability distribution. ...
- 141k
14
votes
Accepted
How to calculate the probability of a data point belonging to a multivariate normal distribution?
Yeah, that sounds right. If you have parameters $\mu$ and $\Sigma$ and data point $x$, then the set of all data points that are less likely than $x$ are the ones that have less density, or in other ...
- 21.5k
14
votes
Accepted
Why do we use parametric distributions instead of empirical distributions?
An enormous amount of data is needed to accurately estimate a distribution nonparametrically, especially a continuous one. Even then, some assumptions about the smoothness of the distribution are ...
- 69.5k
14
votes
Accepted
How can this CDF be decreasing? (power-law)
The vignette by Gillespie (2020) cites Clauset et al (2009) as another source of the figure. Reading that latter paper, it looks like they are actually plotting the complementary CDF (better known as ...
- 133k
14
votes
Accepted
On the proof of right-continuity of the distribution function
First of all, let's write the desired result mathematically: $F$ is right continuous-means really $F(x) = F(x^+)$ for all $x$, where
$$
F(x^{+}) = \lim_{y\to x^+} F(y).
$$
I'm going to assume you are ...
- 12.1k
13
votes
Why is cumulative distribution function monotone non-decreasing?
Because if $x \leq y$, then if $X \leq x$, it follows that $X \leq y$. Therefore, $P(X \leq x) \leq P(X \leq y)$.
More generally, probabilities are monotone in the sense that if $A$ and $B$ are events ...
- 4,107
13
votes
Why is cumulative distribution function monotone non-decreasing?
For a function $f$ to be monotonically non-decreasing, we must have:
$$
f(x+\epsilon)\ge f(x)
$$
for any non-negative $\epsilon$.
Let's check this for the CDF. We have:
$$F(x) = \Pr(X \le x)$$
$$F(x+\...
- 7,752
13
votes
Accepted
Examples of distributions with easily solvable quantile functions but hard to solve CDFs
For a book-length answer to your question, see Statistical Modelling with Quantile Functions by Warren Gilchrist. In the following we assume random variables are continuous, let $F$ the the cdf of $X$,...
12
votes
Accepted
Expectation when cumulative distribution function is given
The discrete case, assume that $X \ge 0$ takes non-negative integer values. Then we can write the expectation as
$$ \DeclareMathOperator{\E}{\mathbb{E}}
\DeclareMathOperator{\P}{\mathbb{P}}
\E X =...
- 82.8k
11
votes
Accepted
Minimum CDF of random variables
Let $x$ by any number. Consider the event $\min(X,Y)\le x$. It can be expressed as the union of two events
$$\min(X,Y)\le x = (X\le x) \cup (Y \le x),$$
shown by the overlapping yellow and green ...
- 334k
11
votes
Accepted
In trouble with CDF graph
Why is it starting from the top-left corner?
The standard* definition of a CDF is
$$
F_X(x) := \mathbb{P}(X \le x)
$$
For reasons which I will never understand, some people plot $S(x) = 1 - F_X(x)$ ...
- 94.1k
Only top scored, non community-wiki answers of a minimum length are eligible