All Questions
31
questions
2
votes
0
answers
69
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Proving $\mathrm E[XY] = E[X] E[Y]$ whenever $X$ and $Y$ are independent random variables.
Let $(\Omega, \mathcal F, \mathbb P)$ be a probability measure space. Let $X$ and $Y$ be two independent random variables on it. Then $\mathrm E [XY] = \mathrm E [X] \mathrm E [Y].$
I tried to do it ...
- 2,612
1
vote
0
answers
128
views
What is the expected length of an interval on an arc of a circle that can be constructed using exponential variates?
Consider a circle $S$ of length $\theta$. Now suppose, we delete an interval $I$ of length $|I|$ (I'll drop the $|\cdot|$ notation for length and directly write $I$) from it.
Now on $S-I$, I choose a ...
- 1,285
0
votes
0
answers
33
views
Calculate the mean number of survivors of a randomly selected insect
An insect lays a large number of eggs $N \sim \operatorname{Poi}(\Lambda), \Lambda>0$. Each of these eggs survives, independently of the other eggs, with probability $p \in (0,1)$. Now suppose we ...
- 762
1
vote
0
answers
31
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Density Function and Expectation
Let $X$ be a real-valued random variable admitting a density whose distribution function $F_X$ is
given by
$F_X(t)=\begin{cases}
0&\text{if}\, t< 2\\
c (t-2)^2&\text{if}\, 2 \leq t \leq 4\...
user1071088
2
votes
2
answers
79
views
Let $Y=h(X),$ Find $E\{Y\}$
Problem:
Let $X: (\Omega, \mathscr{A}) \rightarrow (\mathbb{R},B)$ be a random variable with the uniform distribution $P^X=\frac{1}{2\pi}\mathbb{1}_{\{(0,2\pi)\}}$ on the interval $(0,2\pi),$ and $h:(\...
- 416
-1
votes
1
answer
36
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Classic Hat Problem-Matching problem. Find the expectation of EX_{n} [duplicate]
Problem: At the end of a busy day $n$ fathers arrive at a kindergarten to pick up their kids. Each father picks a child to take home uniformly at random. . Let $X_{n}$ be the matching number among $n$...
- 155
5
votes
1
answer
135
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If $X$ is an integrable random variable, then $\left|\int_{A}X\mathrm{d}\mathbb{P}\right| \leq \int_{A}|X|\mathrm{d}\mathbb{P}$
Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space. If $X$ is an integrable random variable, then
\begin{align*}
\left|\int_{A}X\mathrm{d}\mathbb{P}\right| \leq \int_{A}|X|\mathrm{d}\mathbb{...
- 441
1
vote
1
answer
67
views
$\mathbb{E}[Y|X+Y=z]$
Let $X,Y$ be random variables such that $X\overset{\underset{d}{}}{=}\mathcal{P}\left(\lambda_{1}\right)$ and $Y\overset{\underset{d}{}}{=}\mathcal{P}\left(\lambda_{2}\right)$. Calculate $\mathbb{E}[Y|...
- 991
0
votes
0
answers
91
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Let $X$ be an RV such that $E|X| \lt \infty$. Show that $E|X −c|$ is minimized if we choose $c$ equal to the median of the distribution of $X$.
This question has already answer in mathstackexchange.see here But I am approaching the problem in this way which is not present in mathstackexchange I think.
So Let $X$ be an RV such that $E|X| \lt \...
- 293
0
votes
0
answers
158
views
Proof of 2D-LOTUS in the discrete case
I am self-learning probability theory from Introduction to Probability theory and its applications, by Feller. I would like to ask, if this is a technically correct proof of 2D-LOTUS (Law of the ...
- 5,450
1
vote
1
answer
44
views
Expectation of function of random variable with at most countable range
I am working through Oksendal's SDE book and my answer is slightly different to the solution found online. I was wondering if my solution is correct. This is exercise 2.1(d).
Let $(\Omega, \mathcal{F},...
- 299
1
vote
0
answers
79
views
Why is this probability not sums up to one, wired made up function, please enlighten me.
This is an assignment question, not attempting to fish for a direct solution.
Below is my confusion.
I don't understand why there are $3^3=27$ such functions from $D$ to $\{1,2,3\}$, but each of the ...
- 1,364
0
votes
1
answer
42
views
Expectation of squared error
Let $X \in \mathbb{R}^{m \times n}$ an observation data matrix, $A \in \mathbb{R}^{m \times k}, B \in \mathbb{R}^{k \times n}$ two random variable matrices
I want to calculate:
$$E[(X_{ij} - A_iB_j)^2]...
- 562
2
votes
0
answers
555
views
What is the expected number of consecutive digit pairs “23” in a random integer between 1 and 1,000,000?
had this question on a test and wasn't sure whether my solution is correct. We can use linearity of expectation to look at each two consecutive digits and their expectation for digits 23 as a pair (...
0
votes
2
answers
511
views
Is $\sup_n\mathbf{E}X_n = \mathbf{E}\sup_n X_n$?
I was proving implications of Dominated Convergence Theorem, $\sup_n \mathbf{E}|X_n| < \infty$ if $X_n$ is uniformly integrable, and had this confusion:
If there's a finite collection of uniformly ...
- 19.3k