1
vote
1answer
32 views

Why are linear combinations of independent standard normal random variables also normally distributed?

My professor has given a list of questions that will not be appearing on my test, with this being one of them. I still feel this is extremely important to understand. How can I prove the following ...
0
votes
2answers
59 views

Proving formally $\lim_{x \to -\infty}\mathrm{Pr}( \left \lfloor{x}\right \rfloor \le X < x) = 0$ (Proof check)

we have $$\lim_{x \to -\infty}\mathrm{Pr}( \left \lfloor{x}\right \rfloor \le X < x) $$ where X is a real random variable, and $x \in R$. My idea of a proof would be by contradiction: Assume ...
0
votes
2answers
43 views

Formal Proof: P(A∩B'∩C') = P(A) - P(A∩B) - P(A∩C) + P(A∩B∩C)

I'm trying to prove the following: $\newcommand{\P}{\operatorname{\bf P}}\P(A\cap \overline{B}\cap\overline{C}) = \P(A) - \P(A\cap B) - \P(A\cap C) + \P(A\cap B\cap C)$ I can explain it with a venn ...
2
votes
1answer
64 views

continuous time Markov chain, something the book does not explain

I have a problem with something in my book, under the chapter of continuous time Markov chains. I will post a link to what the book does. They do something which they seem to take for granted, but I ...
0
votes
1answer
60 views

Showing that $E[X|X<x]$ is smaller or equal than $E[X]$ for all x

I would like to show that: $\hspace{2mm} E[X|X<x] \hspace{2mm} \leq \hspace{2mm} E[X] \hspace{2mm} $ for any $x$ X is a continuous R.V. and admits a pdf. I'm guessing this isn't too hard but I ...
0
votes
1answer
186 views

Almost surely equal random variables and expectation

Let $X, Y$ be random variables defined on the same probability space $(\Omega, \mathcal{F}, P)$. I'm interested in seeing a proof for the following results: a) If $P(X = Y) = 1$, then $E(X) = E(Y)$. ...
2
votes
1answer
99 views

Equality of sets when minimizing Shannon's Entropy

Let $P = \{p_1, \ldots, p_n\}$ be a set of probabilities, i.e., $0 \leq p_i \leq 1$. $P$ is such that $\sum_{p_i \in P} p_i = 1$. I have a set of actions $\mathcal{A} = \{a_1, \ldots, a_N\}$ that can ...
1
vote
2answers
98 views

Law of large numbers?

Given random variables $Z_1,Z_2,Z_3,\ldots$, which are uniformly distributed for $[8,10]$: If $X_k =\min\{Z_1, Z_2,Z_3,\ldots,Z_k\}$, prove convergence in probability and find the constant. ...
1
vote
2answers
63 views

Probability Proof.

Write a proof to show that $\mathbb{P}(X_1 \mid X_3) + \mathbb{P}(X_2 \mid X_3) - \mathbb{P}(C_1 \cap X_2 \mid X_3) = \mathbb{P}(X_1 \cup X_2\mid X_3)$ labeling theorems used for each step. My ...
1
vote
1answer
82 views

proof that there is a random variable for which a function has a zero value

Given a function $h$: $$ h(x)=af(x)−b[1−F(x)], $$ where $a$ and $b$ are constants with $b>0$, $f$ is a probability (a generalized) density function and $F$ is its CDF, I want to ...