0
votes
0answers
8 views

Random Variable Problem with unrestricted Parameters Worded Problem

I have no idea how to go about solving (a) -> (c) For (a) Is K=0.2 because k/1-0.8=1 Hence, P(Z=z) = 0.2(0.8)^x But How do we determine the mean or variance with unrestricted z values.
0
votes
1answer
11 views

Expectation of Random Variable - Probability Worded Problem

The part I am confused with is (c) I found part (a) which is: p(0) = 7/24, p(1) = 21/24, p(2) = 7/40 and p(3) = 1/120 How do we find the values for a and b, for part (c) ?
0
votes
1answer
23 views

Expanding the expected value

How to expand: $E(Y+1)^2$ my working out: $E(Y^2)+E(1^2) = E(Y^2)+1$ (I'm not sure why this is though..) Can someone link to or list the rules for expanding the expected value ......
1
vote
2answers
16 views

Finding values of a constant in a probability distribution

A probability distribution for the random variable $X$ is defined by: $$\mathbb{P}[X=x] = K\cdot(0.9)^x,\quad x = 0,1,2,\ldots$$ It is asked to find $\mathbb{P}[X\geq 2]$. When there is a domain for ...
1
vote
1answer
33 views

Expectation of uniform distribution with unknown parameter, given maximal (minimal) observation.

Let $x_i \text{ be} ~ i.i.d. ~ \sim Uni[0,\theta]$ $(\theta \text{ unknown})$. Denote $M_n = \max x_i$. So, through circumferential means, I can show that $E(x_1|M_n) = \frac{n+1}{2n} M_n$. The ...
0
votes
0answers
12 views

Ross probability models questions [on hold]

I am studying for a course and have no professors to talk to live, so I hope some members here can be kind enough to help me. Rather than writing everything out, and splitting it up into different ...
3
votes
2answers
68 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
-2
votes
1answer
33 views

Mean of max vs max of mean

If I have say an $n$ collection of 10 random variables $X_1, \ldots, X_{10}$ (so an $n \times 10$ matrix of values) from some underlying distribution whether Gaussian or uniform, and I calculate ...
3
votes
2answers
47 views

Parity of the sum of consecutive Bernoulli random variables

$\newcommand{\Var}{\operatorname{Var}}$I have $X_1,X_2,\ldots,X_{n+1}$ i.i.d. rv, each $X_i$ is a Bernoulli rv with parameter $p$, i.e. $X_i \in \{0,1\}$, $P(X_i=0)=1-p$ and $P(X_i=1)=p$ with $0 \leq ...
2
votes
1answer
27 views

Invariance Properties of Brownian Motion

I am trying to make sense of the Scaling-Invariance and Time-Inversion properties of Brownian motion by producing a sample path. For the record, I am using the following definitions. Let $B(t)$ be the ...
2
votes
1answer
68 views

Proving that three events are mutually independent

Suppose that: the events $A$ and $B\cap C$ are independent. the events $B$ and $A\cap C$ are independent. the events $C$ and $A\cap B$ are independent. the events $A$ and $B\cup C$ ...
4
votes
2answers
129 views

A counter example of Brownian Motion

Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let $(B_t)$ be a Brownian motion and $U$ be a uniform random variable on ...
0
votes
0answers
27 views

Normal approximation with dependent variables

I have a sequence of $N$ dependent random variables $$y_i = \frac{x_i}{||\vec x||_2} \quad \mathrm{for} \quad \vec x \sim \mathcal N(0,\mathbb{1}_N),$$ where the $x_i$ are the iid elements of $\vec ...
2
votes
1answer
21 views

Filtration from a Brownian Motion

The textbook I am reading defines the filtration induced from a Brownian Motion as follows. Let $\{B(t): t \geq 0\}$ be a Brownian Motion defined on some probability space, then we can define a ...
0
votes
1answer
30 views

Simple conditional probability inequality

I'm reading on some branching process theory in Harris' Theory of Branching Processes and encountered an inequality which looks simple but is eluding me. The full version is a bit complicated to ...
2
votes
3answers
43 views

Finding the expected value of a function of random variables

I'm having troubles with finding marginal density functions and expected values in my probability theory class. I was hoping someone would be able to walk me through the solution to this question (I ...
1
vote
1answer
44 views

Poisson, Gamma distribution example.

Can someone explain me answer for these questions? Suppose customers arrive at a store as a Poisson process with λ = 10 customers per hour. The Poisson process of X ∼ Poisson(λ) the time until k ...
1
vote
0answers
42 views

When can one represent the conditional expectation $E[X|Y]$ as $g(Y)$ with continuous $g$?

Given two random variables $X$ and $Y$ we know that $E[X|Y] = g(Y)$ where $g$ is a Borel function. Is it a good question to ask under which condition there exists a function $g$ which will be ...
4
votes
1answer
50 views

Uniform sampling with replacement item frequency

Suppose we are sampling from $N$ distinct items uniformly with replacement $M$ times. What can be said about the distribution of frequencies of items drawn? For example, if I sort all the frequencies ...
1
vote
0answers
49 views

Is my interpretation of Bayesian probability and inference correct?

I have the following interpretation of the Bayesian probability and inference (without referring to Measure Theory, I am still at the very beginning of learning it): Let's say we have five random ...
-3
votes
1answer
33 views

Derive/ prove: p(a,b|c) = p(a|b,c).p(b|c)

How can this expression be derived? p(a,b|c) = p(a|b,c).p(b|c) where a,b,c are random variables. UPDATE: from the following ...
0
votes
2answers
35 views

Finding independence of two random variables

We're learning about independent random variables in the context of multivariate probability distributions and I just need some help with this one question. If $f(y_1, y_2)=6y_1^2y_2$ when $0\leq y_1 ...
1
vote
1answer
40 views

Proof of “continuity from above” and “continuity from below” from the axioms of probability

One of the consequences of the axioms of probability ($\sigma$ field and probability axiom) is the "infinite subset" and "infinite union" property, I can't figure out how it follows from them. if ...
2
votes
1answer
51 views

$\mathbb E[\mathbb E(X|Y, Z)|Y]$ or $\mathbb E\{\mathbb E[(X|Y)|Z]\}$?

To begin with, the standard iterated law of probability is as follows. $$ \mathbb E X = \mathbb E [\mathbb E(X|Y)]. (1) $$ I am perfectly happy with $(1)$ and there is also some quite good ...
1
vote
2answers
60 views

Proof of, and requirements for, the reverse of Jensen's Inequality for concave functions

As I understand it, Jensen's Inequality states $$\int_{U}f_{V}\left(h(u)g(u)\right)du\geq f_{V}\left(\int_{U}h(u)g(u)du\right)$$ For a convex function $f_{V}$, a probability distribution $g(u)$ on ...
-2
votes
0answers
38 views

Characteristic functions

Here $E(Y)$ means the expected value of $Y$. 1) Could any one explain for me how to get from (2.7) to (2.8) ? 2) Why does the author know to define $\phi_1(u)$ and $\phi_2(u)$ in such a way? ...
0
votes
0answers
50 views

Writing probability as log

I have a question regarding the log probability and I am confused on this. The question is: $$\hat P^{(t)}(x)=\sum_{i=1}^N v_i^{(t)}P_i^{(t)}(x)$$ which is some function of size $N$. The question ...
0
votes
1answer
23 views

Total law of probability in continuous space

I am finding little difficulty in the following definition of total probability specified in a NLP related paper. Say $q^i$ is a partition of my continuous sample space. The authors have defined the ...
0
votes
0answers
13 views

Stopped supremum of the Brownian local time still $L^p$ bounded in space?

Let $B_t$ be a standard Brownian motion and $L_t^x$ its local time in $x$ at time $t$. For fixed $t$ and $p>1$, it holds that $$ \sup_{x \in \mathbb{R}} \operatorname{E} [ (L_t^x)^p ] < ...
0
votes
0answers
29 views

Probability Distributions and Random Discrete Variables

How do you read this? For (a) do we let $X= 1/6, 1/2, 1/5$ and $2/15$ and sub into the equation, $$ Y=X^2-2X. $$ How do we go about solving this?
2
votes
1answer
31 views

Difference between $\lim P[…]$ and $P[ \lim ]$

In a Galton-Watson branching process the extinction probability is sometimes given by $$\lim_{t \rightarrow \infty} P[X(t)=0]$$ and sometimes as $$ P[\lim_{t \rightarrow \infty}X(t)=0]$$ Is there a ...
0
votes
1answer
23 views

Conditional Probability using a Matrix

I understand how to find P1: that is simply: P(D1|D0)=0.8 P(W1|D0)=0.2 P(D1|W1)=0.4 P(W1|W0)=0.6 I do not however, understand how to find P2 using the matrix. Normally I would solve it as ...
2
votes
1answer
49 views

Relation between uniformly distributed random variable and i.i.d Bernoulli sequence (Cantor space)

(Uniform RV <==> i.i.d Bernoulli sequence) (1) Let $(X_n)_n$ be a sequence of i.i.d. Bernoulli random variables($P(X_n=0)=P(X_n=1)=\frac 12$) on a probability space. Then show that $\xi:= \sum_n ...
1
vote
2answers
18 views

Proving the Probability of an Event Through Bayes Theorem.

The question goes as such: An event A can occur if only one of the mutually exclusive events B1, B2, or B3 occur. Show that P(A) = P(B1)P(A|B1)+P(B2)(A|B2)+P(B3)*(A|B3) my working out: P[A|(B1 U B2 ...
0
votes
0answers
36 views

Sampling and averaging in Monte Carlo Simulation

(First of all, I apologize for the vague title. Couldn't think of rather proper one.) Let's say that we have 10 items where each item has probability distribution of one's own, say Lognormal ...
0
votes
1answer
45 views

solving a simple inverse problem related to elliptic pde

Suppose that I have the elliptic PDE $\nabla(\nabla A(x)\cdot U(x)) = 0$ where $x \in [0,l_1]\times [0,l_2]$ with boundary conditions $U(0,x_2) = 0, U(l_1,x_2)=1$ and $U_{x_1}(x_1,0)=0, ...
1
vote
2answers
29 views

Finding the mean and variance of an exponential probability distribution

I'm taking a probability theory course, and I'm struggling a bit with gamma and exponential distributions. Here's a question that I'm stuck on: The length of time Y necessary to complete a key ...
0
votes
1answer
30 views

How to check hypothesis in statistical data?

I have a statistical problem. In a city there are some hostels which differ by the number of rooms. The input data are the following. In a table there is information about hostels and corresponding ...
0
votes
0answers
21 views

Secretary problem *without* each ordering equally likely

Also known as: the marriage problem, the sultan's dowry problem, the fussy suitor problem, the googol game, and the best choice problem. See http://en.wikipedia.org/wiki/Secretary_problem $n/e$ is ...
0
votes
1answer
24 views

(Multidimensional) Standard Brownian Motion: Convergence

Relating to this question, I have a further one, and hope, someone can help me. I know that $$\left(X_j - X_{j-1}\right)_{j=1}^t \xrightarrow{d} \left(Y_j\right)_{j=1}^t.$$ Further, we know that ...
0
votes
1answer
36 views

Regular dependence

I have seen the definition of "regular dependence" in many books (usually old books), but I could not fully understand that definition, hope you can help me understand it. The dependence of $X$ and ...
0
votes
1answer
46 views

Why do we need to use random variables

In my statistics textbook (The Practice of Statistics by Starnes, Yates, and Moore) an example is given. In it, 21 students are each given three glasses of water. Two are filled with tap water and one ...
3
votes
0answers
27 views

Itô Excursion Measure

I am looking for any source of information regarding Itô Excursion Measure (for Brownian Motion). I am looking for a selfcointained reference (Though I have basic knowledge on Local Times and Poisson ...
0
votes
2answers
58 views

Probability distribution of $\min(X,Y)$ given that $\max(X,Y)>1/2$ [closed]

Suppose $X$ and $Y$ are two independent random variables. What is the value of $\Pr[\min(X,Y) \leq z \mid \max(X,Y) >1/2]$? They both follow a Uniform distribution with parameters 0 and 1
0
votes
0answers
27 views

Bayes Theorem with multiple observations

Let $H \in \{1,..,K\}$ be a discrete random variable and $e_1, e_2$ be observed values of 2 other random variable $E_1$ and $E_2$. We wish to calculate the vector ...
2
votes
2answers
59 views

If $ P(A) = 0 $ is $ A $ a null event?

I know that $ P(\text{null event}) = 0 $, but is the reverse true? i.e. if $ P(A) = 0 $ is $ A $ a null event? I'm not too sure I even understand what a null event is, to be honest. Could anyone give ...
1
vote
1answer
40 views

If $f$ is a pdf can we construct $g$ such that $x\sim U[0,1)$ implies $g(x)\sim f$

Let $f$ be some pdf over $[0,1)$. Here is my question: does there always exist an infinite partition $\{X_{s}\}_{s\,\in\, \mathrm{support}(f)}$ of $[0,1)$ such that if we define $g(x):[0,1)\rightarrow ...
0
votes
0answers
27 views

Markov chain with Poisson distribution

Let $X \in \mathbb{R}^+$ and $Y \in \mathbb{Z}^+$ be the Random Variables (RVs) where the condition PDF $f_{Y|X}(y|x)$ follows a Poisson distribution as $$ f_{Y|X}(y|x) = ...
4
votes
1answer
121 views
+50

Unbiased asymptotic variance

Problem: Let $X_1,...,X_n$ be indep. r.v.'s that satisfy, for $i = 1,...,n$, $E(X_i) = \mu_i(\theta)$ & $\mathrm{Var}(X_i)= \sigma_i^2(\theta)$. $\theta$ is the parameter of interest and the ...
0
votes
2answers
43 views

Existence of density function for a sum of 2 Random Variables

Let's suppose that $Y$ is the normal distribution and that $X$ is another random variable whose density function may or may not exist. Does it follow that $Y+X$ has a density function? I am reading ...