438 views

### How to interpret sum of two random variables that cross domains?

suppose we have two discrete random variables: $X: \{$6 sided dice rolls$\}$ $\rightarrow \{1..6\}$ (following uniform distribution) $Y: \{$coin flips$\}$ $\rightarrow \{0,1\}$ (following uniform ...
26k views

### Why do we need sigma-algebras to define probability spaces?

We have a random experiment with different outcomes forming the sample space $\Omega,$ on which we look with interest at certain patterns, called events $\mathscr{F}.$ Sigma-algebras (or sigma-fields) ...
36k views

### Why is the sum of two random variables a convolution?

For long time I did not understand why the "sum" of two random variables is their convolution, whereas a mixture density function sum of $f(x)$ and $g(x)$ is $p\,f(x)+(1-p)g(x)$; the arithmetic sum ...
120 views

### why can two random variables be added only when they have the same domain?

I am watching lecture 7 in harvard stats 110 and the professor is teaching distribution of addition of two random variables and in a breadth says that random variables can be added only if their ...
66 views

### What does it mean to obtain a sample $S$ of size $n$ according to a distribution $D$ over a set $X$ in machine learning?

What does it mean to obtain a sample $S$ of size $n$ according to a distribution $D$ over a set $X$ in machine learning?
139 views

### Sum of Discrete Random Variables [duplicate]

If I have two independent discrete random variables, say, $$X \in \{1,3,10,20\}$$ and $$Y \in \{2,3,5,9,11,15\}$$ and let $$Z = X + Y$$ be the sum of two variables. Also, each value taken by ...
184 views

### what exactly does it mean when we say “Let $X_1, X_2 …$ be iid random variables”

Every now and then I read that phrase and get confused. When we say "Let $X_1, X_2, \dots X_n$ be iid random variables" I thought this meant that we are sampling $X$ random variable n many times ...
17k views

### Empirical CDF vs CDF

I'm learning about the Empirical Cumulative Distribution Function. But I still don't understand Why is it called 'Empirical'? Is there any difference between Empirical CDF and CDF?
12k views

### What is meant by a “random variable”?

What do they mean when they say "random variable"?
93 views

### Is $\bar X$ a random variable or a constant?

I am confused how $\bar X$ is used sometimes as a constant and othertimes as a random variable. My understanding is that $\bar X$ is a random variable because it changes every time our sample changes....
94 views

### Does a pair of random variables $X$ and $Y$ form a new sample space?

I am studying probability theory from Wasserman's All of Statistics. The author does not mention the concept of a probability space, although he does mention all of its components separately. Please ...
181 views

### I can't understand the definition of the convergence in probability

https://en.wikipedia.org/wiki/Convergence_of_random_variables Wikipedia defines convergence in probability as follows: A sequence $X_n$ of random variables converges in probability towards the ...
239 views

### How to add and multiply distributions?

I saw in a statistics book a problem. Let $X$ be a distribution that gets $1$ for probability $0.4$ and $2$ for probability $0.6$. Compute the mean and variances of $Y=3X-2$ and $Y=3X^2-2$. I found ...