# Proving a self independent random variable can get only one value

I’m doing a course in probability and was asked the prove the following. I’d appreciate some feedback on it, thank you.

Let $$X$$ be a self independent r.v. . Prove that exists $$cin mathbb{R}$$ s.t. $$mathbb{P}(X=c)=1$$

My proof:

Let A denote the support of $$X$$.

Lets assume that exists $$x,yin A$$ such that $$xneq y$$.

Let $$B_1= {x}$$ and $$B_2={y}$$.

So we get:
$$mathbb{P}(Xin B_1 cap B_2) = mathbb{P}(Xin B_1, ;Xin B_2) = mathbb{P}(Xin B_1) cdot mathbb{P}(Xin B_2)=q_1 cdot q_2 >0$$

(this is true because $$x,y in A$$ and have a non zero probability and $$X$$ is self independent).

But $$B_1 cap B_2 = emptyset Rightarrow mathbb{P}(X in B_1 cap B_2)=0$$ – Contradiction!

And since the support of a r.v. can not be empty we get that $$A={x}$$ and by definition

$$mathbb{P}(X=x)=1$$

Edit: Thank you for your comments, I’ll fix it

Mathematics Asked by override on November 12, 2021

1 Answers

## One Answer

By "self independent" I presume you mean $$X$$ and $$X$$ are independent.

Suggestion: Rather than looking at $$P(X = x)$$ which might always be $$0$$, consider the cumulative distribution function $$F(x) = P(X le x)$$. Show that this is always $$0$$ or $$1$$.

Answered by Robert Israel on November 12, 2021

## Related Questions

### Let $ABCD$ be a cyclic quadrilateral and let $AB$ and $CD$ meet at $E$. Let $M= (EBC)cap (EAD)$. Prove that $OMperp EM$

1  Asked on December 16, 2020 by raheel

### Computing the limit of an integral of a function series

2  Asked on December 16, 2020 by e2r0ns

### Show that area are equals

0  Asked on December 16, 2020 by kevin-duran

### Using scaled equations to go from $rho u_{tt}(x, t) + Ek^2u_{xxxx}(x, t) = 0$ to $v_{tau tau} + Jv_{zeta zeta zeta zeta} = 0$

0  Asked on December 16, 2020 by the-pointer

### Is a generated field independent of the extension over which it is generated?

1  Asked on December 16, 2020 by jam

### Limit of hypergeometric distribution when sample size grows with population size

2  Asked on December 16, 2020 by tc1729

### Arg of $(1-isqrt{3})^6$. Did I do it right?

3  Asked on December 16, 2020 by cocacola

### Is there closed formula to calculate the probability of beta distribution?

1  Asked on December 16, 2020 by vesii

### Coloring Two Faces of an Icosahedron

2  Asked on December 16, 2020 by user826216

### prove that there is only one a, such: $y'(x)=y(x)^3+sin(x);y(0)=a$ and y(x) is a unique periodic solution

0  Asked on December 16, 2020 by user819065

### Proof that there are only two automorphisms of $mathbb{Q}(sqrt d)$ fixing $mathbb{Q}$

1  Asked on December 15, 2020 by hyperpro

### Find $lim_{xto 0} frac{sqrt{ax+b}-1}{x}=1$

3  Asked on December 15, 2020

### Improper integral of unbounded function over bounded interval

1  Asked on December 15, 2020 by wykk

### representing of expectation for submatrices

0  Asked on December 15, 2020 by user4164

### A boundary condition

0  Asked on December 15, 2020 by lorenzo-andreaus

### Solving the Fractional Fourier Integral Transform of $e^{j omega_{0} t}$

0  Asked on December 15, 2020 by the-dude

### A complete bipartite graph is unique

1  Asked on December 15, 2020 by itsnotme

### Global minimum for $frac{2(q – 1)(q^k + 1)}{q^{k+1} + q – 1}$, if $q geq 5$ and $k geq 1$

1  Asked on December 15, 2020 by arnie-bebita-dris

### Behaviour of orthogonal matrices

2  Asked on December 15, 2020 by a9302c

### A single reference of Real Analysis/Calculus with following content

0  Asked on December 15, 2020 by beginner

### Ask a Question

Get help from others!

© 2021 InsideDarkWeb.com. All rights reserved.