Modern theory of probability is formulated on the footing of measure theory. Use this tag if your question is about this theoretical footing (for example probability spaces, random variables, law of large numbers, central limit theorems, and the like). Use (probability) for explicit computation of ...

learn more… | top users | synonyms (1)

0
votes
1answer
33 views

Question on Martingales and Brownian Motion

I've run into an issue and I am unsure of how to proceed. In the text I'm working through, the following is left "as an exercise to the reader." Normally proofs listed as such tend to be fairly simple ...
0
votes
1answer
41 views

$$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& elsewhere.\end{cases}$$ Find the MLE for $θ$.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
1
vote
2answers
27 views

Finding correlation coefficient

Two dice are thrown. $X$ denotes number on first die and $Y$ denotes maximum of the numbers on the two dice. Compute the correlation coefficient.
2
votes
1answer
52 views

Is the graph of a Brownian motion over an interval measurable?

Let $n \in \mathbb{N}_1 := \{1, 2, \dots\}$ and let $B:\Omega \times [0, \infty) \rightarrow \mathbb{R}^n$ be a standard, $n$-dimensional Brownian motion over the probability space $(\Omega, ...
0
votes
1answer
26 views

Show that if for $n \ge 1$, $(X_0, \ldots, X_n)$ has probability function $f_n$, then $X$ is a Markov Chain with transition matrix $\mathbb P$

Let $S$ a set of states and $\mathbb P=\{p_{i,j}\}_{i,j \in S}$ be a transition matrix. I've proved that $f_n(i_0,\ldots,i_n) := f_0(i_0) p_{i_0 i_1} \dots p_{i_{n-1} i_n}, \ \ (i_0, \ldots, i_n) ...
0
votes
1answer
25 views

Find $E[VY]$ where V is a continuous RV and $Y$ is a discrete RV.

Let $V \sim R(0,1)$ be uniform continuous and $Y$ discrete with density: $\frac 1 4$ if $y=0$, $\frac 3 4$ if $y=1$. Find $E[VY]:$ I know $P(VY = 0)=\frac 1 4$ and $P(0<VY\le 1) = \frac 3 4$. ...
1
vote
2answers
34 views

$$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& elsewhere.\end{cases}$$ Find an estimator for $θ$ by the method of moments.

Let $Y_1$, $Y_2$, . . . , $Y_n$ denote a random sample from the probability density function $$f (y | θ)=\begin{cases} (θ + 1)y^θ, & 0 < y < 1; θ > −1,\\ 0 ,& ...
1
vote
1answer
28 views

Show that for $0<k<1$ $P(k < \frac{Y_{(n)}}{\theta} \le 1) = 1 - k^{cn}$.

The distribution function for a power family distribution is given by $$F(y)=\begin{cases} 0, & y<0\\ \left(\frac{y}{\theta}\right)^\alpha, &0\le y \le \theta \\ 1, ...
0
votes
2answers
23 views

Probability of 3 dices

Been looking through past exam papers and came across this question: Three fair dices are rolled. The probability that all three dices show 5 is 1/216. Is this true?
0
votes
0answers
11 views

Concentration bound for $f(w) = w \times \sin wz$

I need to find an exponential bound for $P(|S_n - \mu| > \lambda)$ where $S_n = \frac{1}{D} \sum_{i=1}^D w_i \sin w_iz$ for a constant $z$, $E(S_n) = \mu$ and $w_i$ are drawn from the normal ...
0
votes
0answers
9 views

Derive concentration bound for the derivative

It that true to conclude that if a random $f(z)$ is a sub-Gaussian random variable for a constant value of z, its derivative $f'(z)|_{z=k}$ with respect to variable $z$ is also sub-Gaussian? In ...
0
votes
1answer
29 views

Possible outcome

A bag containing 6 white & 6 black socks . A sock is drawn from the bag, and it's color recorded, & put back in the bag. This is done 3 times. Part A : show all possible outcomes. Let w ...
1
vote
0answers
37 views

Delta Function as a Conditional Distribution

This is problem 20 from chapter 21 of A Modern Approach to Probability Theory by Fristedt and Gray: Suppose that $X$ is a random variable measurable with respect to a $\sigma$-field $\mathcal{G}$. ...
0
votes
0answers
10 views

A Question on the independence of the sample mean and sample variance

The aim of the following question is to show the given random variable follows a student T distribution. Although it seems quite straightforward at the first sight, I am quite confused about the ...
0
votes
1answer
33 views

number of ways to put n balls in n-1 slots so that only 1 has 2 balls

The problem says We have n balls and n slots. And asks for the probability that exactly the first slot has no balls and exactly one of them has 2 balls. My attempt: There are $n^{n}$ total ...
2
votes
1answer
23 views

Usefulness of criterion for weak convergence

I am currently reading the book Convergence of Probability Measures by Patrick Billingsley, and I came across the following theorem: Theorem. Let $(S,\rho)$ be a metric space, and $B(S)$ be the Borel ...
0
votes
0answers
15 views

Monte carlo formula to compute the approximation of variance of MLE

In the book of "Monte Carlo Statistical Methods", the book gives an approximation formula for the variance of MLE, Later on, the book mentions that this approximation formula can be written as ...
1
vote
3answers
69 views

Let $N_n$ be the number of throws before all $n$ dice have shown $6$. Set $m_n := E[N_n]$. Write a recursive formula for $m_n$.

Suppose we throw $n$ independent dices. After each throw we put aside the dices showing $6$ and then perform a new throw with the dices not showing $6$, repeating the process until all dices have ...
1
vote
1answer
35 views

A basic question on Lindeberg and Lyapunov condition

Suppose along with independence in each row of a triangular array, it is given that in each row random variables are identically distributed then what Lindeberg and Lyapunov condition reduces to ? ...
0
votes
0answers
28 views

Stationary distribution of a “birth-death model” that does not have Markov property

A typical birth-death process is defined such as the probability of going from any state $j$ to any state $i$ is given by: $$ p_{ij}= \begin{cases} b_i & \text {if $j = i+1$} \\ ...
1
vote
2answers
22 views

$X_i \sim e(\lambda_i), i = 1,2$ be independent. Show $P(X_1 > X_2) = \frac {\lambda_2}{\lambda_1 + \lambda_2}$ using the Law of Total Probability.

Let $X_1,X_2$ be independent, $X_i \sim e(\lambda_i)$ for $i=1,2$. Find $G(x_2) = P(X_1 > x_2): \int^{\infty}_{x_2} \lambda_1 e^{-\lambda_1 x_1} \ d_{x_1} = e^{-\lambda_1 x_2}$ Now show $P(X_1 ...
2
votes
1answer
34 views

Let $X,Y$ be independent RV's and suppose $F_X$ is continuous. Show $P(X=Y)=0$ and tell if $P(X=Y)=1$ is possible if $F_X$ is not continuous.

Let $X,Y$ be independent RV's and suppose $F_X$ is continuous. Show $P(X=Y)=0$ and tell if $P(X=Y)=1$ is possible if $F_X$ is not continuous. My idea is: $P(X=Y) = F_X(Y=y) - \lim_{z \rightarrow ...
1
vote
1answer
34 views

Compute $E[X \mid Y = y], EX$ and $E[XY^2]$.

Compute $E[X \mid Y = y], EX$ and $E[XY^2]$. I know $X \mid Y = y \sim ye^{-yx}$ and $Y \sim e(1)$ and $x,y >0$. To compute I do as follows: $E[X \mid Y = y] = \int^{\infty}_0 x\cdot ...
2
votes
3answers
51 views

What is the probability that given a bin (out of $n$ bins) that it will receive no balls

I'm working on a problem where there are $n$ bins. I want to know the probability that given a bin (Specific bin) that it will receive zero balls. There are $m$ balls thrown at random into the bins. ...
2
votes
1answer
19 views

Mean of two normal variates

If $X$ and $Y$ are independent standard normal variates find the mean value of the greater of $|X|$ and $|Y|$
-3
votes
0answers
32 views

Grad level probability

Thirty six of the staff of 80 teachers at a local school are certified in CPR. In 180 days of school, about how many days can we expect a the teacher on bus duty will likely be certified in CPR?
0
votes
1answer
20 views

interacting probabilitys

Find two absolutely continuous probability measures $\mu(x)dx$ and$\nu(x)dx$ with finite second moments. Such that the function $f(t)$ we have that $\dfrac{d^{2}}{dt^{2}}f(t)<0$ where ...
0
votes
1answer
29 views

Ten children standing in line

Ten children (five boys and five girls) are standing in line. Assume that all possible ways in which they might line up are equally likely. What is the probability that between any two girls there ...
1
vote
0answers
13 views

Union of binary cylinder sets

Suppose, $S=\{0,1\}^\infty$ and say $\Sigma_0 \subset 2^S$, such that $$\Sigma_0=\{\{s | s_{i_k}= a_{i_k}, 1 \leq k \leq n, a_{i_k} \in \{0,1\}\}, n \in \mathbb{N}, i_1 \leq i_2 \leq \cdots \leq ...
0
votes
1answer
35 views

D6 Event Tree Probability Question. [closed]

One has tried looking this one up and Googling it, One is also dispraxic so while math can be tricky if one can get the concept and explanation behind something generally work at it until one ...
0
votes
4answers
24 views

Question regarding inequalities and probability.

I was doing a problem involving standard normal random variables where the solution involved this particular step: $P\{a<X<b\}=P\{X<b\}-P\{X\le a\}$ I haven't been able to find out as to why ...
0
votes
2answers
25 views

What is Poisson Point Process?

"Points $\{A_j\}_{j\in\Phi(\lambda)}$ are assumed to be distributed according to a homogeneous PPP with intensity $\lambda$, denoted $\Phi(\lambda)=\{X_j\}$, where $X_j$ is the location of the $j$th ...
1
vote
0answers
27 views

Moments of stable random variables [closed]

I read in Handbook of Heavy Tailed Distributions in Finance that it is a 'well known fact' that: The $p$th absolute moment of a symmetric stable random variable (with index $\alpha \in (0,2) $) is ...
0
votes
0answers
19 views

Discrete Markov Chain: probabilities

I'm confused about these: steady-state transition probabilities limiting probabilities stationary probabilities how are they different? I know the question is pretty vague, but I feel like I'm ...
1
vote
2answers
23 views

$K$ events that are $(K-1)$-wise Independent but not Mutually/Fully Independent

I had the following question: Construct a probability space $(\Omega,P)$ and $k$ events, each with probability $\frac12$, that are $(k-1)$-wise, but not fully independent. Make the sample space as ...
0
votes
0answers
21 views

Identification of linear regression function under $\ell_1$-norm criterion.

Consider a linear system \begin{align*} y = \theta^Tx + e, \end{align*} where $x\in\mathbb{R}^n$ and $e\in\mathbb{R}$ are independent Gaussian random variables with distribution $\mathcal{N}(0,I_n)$ ...
2
votes
1answer
34 views

About Lévy metric

From Wikipedia: Let $F, G : \mathbb{R} \to [0, 1]$ be two cumulative distribution functions. Define the Lévy distance between them to be :$$L(F, G) := \inf \{ \varepsilon > 0 | F(x - ...
0
votes
0answers
49 views
+50

Derive minimum length confidence bounds for a F distribution variance …

Derive minimum length confidence bounds for a F distribution variance $\sigma^2$ and the ratio of two F distribution population variances $\frac{\sigma_1^2}{\sigma_2^2}$. What I got so far is $$ ...
1
vote
0answers
12 views

Does $(X⊥Y | z^0)$ for binary-random variable Z implies $(X⊥Y | Z)$?

I have been attempting to reason about this question. Does $(X⊥Y | z^0)$ for binary-random variable Z implies $(X⊥Y | Z)$ ? My intuitive thought is that this is not possible but I would really ...
-2
votes
1answer
36 views

Event-level independence does not imply random-variable independence for non-binary random variables

A single event-level independence $(x^0⊥y^0)$ does not imply the random variable level independence $X⊥Y$ for non-binary random variables. By non-binary I mean, random variables has more than two ...
1
vote
3answers
34 views

Independence of max and min of a set of random variables.

Suppose $X_1,\ldots,X_n$ are independent and identically distributed random variables with cdf $F_X(x)$. Define $U$ and $L$ as $U=\max\{ X_1, \ldots ,X_n\}$ and $L = \min\{X_1,\ldots,X_n\}$. Are $U$ ...
1
vote
1answer
42 views

Convergence to integral: $\sum_{k=1}^{k_n}f\left(B_{t_{k-1}^{(n)}}\right)\left(B_{t_{k-1}^{(n)}}-B_{t_{k}^{(n)}}\right)^2 \to_p \int_0^Tf(B_t)dt$

The problem goes: Let $(B_t)$ be a standard Brownian motion, and $f:\mathbb{R}\to\mathbb{R}$ be continuous. Show that if $T>0$ and $(P_n)$ is a sequence of partitions of $[0,T]$: ...
1
vote
1answer
13 views

Probability using Chebyshev equation

A dice is rolled until the result is 6 and this is repeated n times. Sn represents the total amount of times the dice is being rolled (in order to get 6 total of n times). Find the probability for ...
0
votes
1answer
24 views

Show if $X,Y$ are independent RV's (discrete or continuous) then $k+X,Y$ and $kX,Y$ are indepedent for $k \in \mathbb R$.

Show if $X,Y$ are independent RV's (discrete or continuous) then $k+X,Y$ and $kX,Y$ are indepedent for $k \in \mathbb R$. I've been thinking how to prove the above statement. Intuitivelly it's ...
0
votes
2answers
19 views

probability related to uncountable set

Let $X$ be the collection of closed interval of the form $[a,1],$ where $a \in [0,1]$ and we fix a real number $t \in [0,1]$. Suppose an element $c$ is randomly drawn from $X$, what is the probability ...
0
votes
0answers
21 views

probability distributions [closed]

a regional visits local kfc franchise outlets and evaluates the speed of service . if a customer receives his/her meal within 45 seconds, the server is given a free movie-admission ticket. if it takes ...
0
votes
1answer
32 views

Let $(X,Y)$ be an absolute continuous R.V. Find $XY \mid Y = y$ and show $XY$ and $Y$ are independent. Also $XY \sim e(1)$.

Let $(X,Y)$ be an absolute continuous R.V with density $f_{X,Y}(x,y) = ye^{-y(x+1)}, \ x,y >0$. I've shown that $Y \sim e(1)$ and $X \mid Y = y$ density $x \mapsto ye^{-yx}$. However I must find ...
0
votes
1answer
22 views

Product of two random variables together with conditional density

Let $X_1$ and $X_2$ be two real valued random variables such that we have the conditional density of $X_1$ given $X_2$, i.e. $$\mathbb P(X_1\in M\mid X_2) = \int_M \phi(x_1\mid X_2)dx_1$$Also, let $h$ ...
0
votes
1answer
47 views

Stopping time question $\sigma$

If $S$ and $T$ are stopping time, $S \vee T$ is $\max ({S,T})$, $F_S$ and $F_T$ are stopped sigma algebra, show that $F_{S \vee T} = \sigma(F_S,F_T)$. My thinking : I should take a set $A$ in $F_{S ...
2
votes
1answer
29 views

Show that $\dfrac{\max(X_1,\cdots,X_n)}{n}\xrightarrow{a.s} \,0$.

Let $\{X_n\}$ be a sequence of identically distributed random variables with $\text{E}(|X_1|)< \infty$. Show that $\dfrac{\max(X_1,\cdots,X_n)}{n}\xrightarrow{a.s} 0$ I need to show ...