3
votes
1answer
21 views

Convergence of expected values as random variables converge almost surely

Let I have a sequence of random variables $X_n$ that converges to random variable $X$ almost surely as $n\to\infty$. How can I proof that $\lim_{n\to\infty}\mathcal{E}[X_n]=\mathcal{E}[X]$ where ...
1
vote
1answer
32 views

Can this function be a density function of a continuous random variable X?

F(x) = 0, if x < 1 F(x) = 1, if 1<=x<=2 F(x) = 0, if x>2 I think it could be, as long as the integral is 1. Any ideas?
0
votes
1answer
14 views

How to calculate the median and the quantiles of this distribution?

A median of a distribution of one random variable $X$ of the discrete or continuous type is a value of $x$ such that $P(X < x) \leq 1/2$ and $P(X \leq x) \geq 1/2$. If there is only one such $x$, ...
1
vote
0answers
24 views

Sample Mean and Sample Variance

Consider a sample of data S obtained by flipping a coin x, where 0 denotes the coin turned up heads, and 1 denotes that it turned up tails. S = {1, 1, 0, 1, 0} What is the sample mean for this data ? ...
1
vote
2answers
13 views

Geometric and binomial distribution problem

Let $X \sim Bi(n,p)$, and $Y \sim \mathcal{G}(p)$. (a) Show that $P(X=0)=P(Y>n)$. (b) Find the number of kids a marriage should have so as the probability of having at least one boy is $\geq ...
1
vote
0answers
30 views

Finding joint distribution function?

Let $U$ and $V$ be two independent uniform (0,1) random variables and let \begin{eqnarray*} R &=&\sqrt{\frac{U^{2}+V^{2}}{2}},\\ H &=&\frac{2U}{U+V}. \end{eqnarray*} I found the ...
-1
votes
1answer
27 views

Probability Question for two head coin [on hold]

I have tried to find the solution of this question but i am not able to get the correct solution for this... It will be very beneficial for me if someone provides me the complete solution to this ...
0
votes
1answer
35 views

Finding a probability density function of a function of three dependent random variables

I have three random variables that are functions of another three random variables by pairs, say: $U=fc(X,Y)$, $V=fc(Y,Z)$ and $W=fc(X,Z)$, with $X$, $Y$ and $Z$ being independent random variables ...
0
votes
0answers
52 views

Upper bound on the covariance of two gamma processes?

Given two binary gamma processes, $X = \Gamma(t; \gamma_1, \lambda_1)$ and $Y = \Gamma(t; \gamma_2, \lambda_2)$, what is their maximum covariance? Applying this answer, it would seem that it is the ...
0
votes
3answers
25 views

Median Value + Mode for Hybrid Functions of a Continuous Probability Density Function

To find the median: should I set the integral to 0.5.... but because there are two functions that are non-zero, I am unaware of a method to find the median. To find the mode: would I need to ...
1
vote
1answer
49 views

Sum of random variable

Considering two continuous random variables $X$ and $Y$ with $d.f \; F_X, F_Y$ I want to fin the distribution and distribution function of the sum $Z=X+Y$. \begin{align} P\{Z \leq z\} &= P\{X+Y ...
0
votes
2answers
28 views

Calculating Probabilities for a cumulative distribution function within a given inequality

Given that K = 1/36, I require some help understanding (b) • Pr(1/2 ≤ X ≤ 1) Is re-written as such: Pr(X ≤ 1) - Pr(X < 1/2) I do not understand why! Is it because Pr(X ≤ 1) is solved as ...
1
vote
1answer
20 views

Calculating Probabilities using a cumulative distribution function

For (b) Pr(X greater than or equal to 2) = ? The textbook says as such but I am confused: Pr(X greater than or equal to 2) = 1 - pr(X less than 2) I do not understand why they re-write the ...
-1
votes
0answers
11 views

Product of Gaussian random variable with hermition of another independent gaussian random variable. [closed]

If X∼ CN(0,1) and Y ∼ CN(0,1). X and Y are vectors independent of one another. How to find the E[(X†)Y]. What will be the probability density funtion of Z, If Z = (X†).Y ?
0
votes
2answers
27 views

I'm not sure if I'm supposed to use a Poisson distribution or Conditional Probability (or both) to answer this question

I have a question that I'm trying to solve. I have the answer but I don't know how they arrived at the answer so I can't compare my work and see where I went wrong. The number of injury claims per ...
0
votes
2answers
11 views

Expected Profit for Binomial Variable

Part (a) I am familiar with: (a) P(batch is rejected) = P(X greater than or equal to 3) and n = 15 and p(defective) = 0.1 This gives me the correct answer of 0.1841 I am stuck at part 2! I have ...
0
votes
2answers
30 views

The relation of $P(X=x+1)$ and $P(X=x)$ in binomial distribution

If I substitute the values to the binomial probability theory, it appears as such $${n \choose x+1} p^{x+1} (1-p)^{n-x-1}$$ I don't know how to move on... What am I doing wrong, or are you ...
0
votes
0answers
22 views

Let $X_t$ be a Brownian motion find $X_2>2, x_1>x_2,$ and $x_t<4$ for all $2\leq t\leq 3 $ [closed]

Let $X_t$ be a Brownian motion find $X_2>2, x_1>x_2,$ and $x_t<4$ for all $2\leq t\leq 3 $ Can you help me with tips and bibliography... I don't understand very good the topic, and I can't ...
0
votes
1answer
31 views

Transformation of random variable

I want to prove the following: $$\text{Let F be a distribution function of any random variable $\\$ and G(x) the quantile function (or inverse) of } \frac 1 {1-F(x)}$$ $$\text{Then, for a standard ...
-2
votes
0answers
12 views

probability distributions [closed]

![ Question 1. 1. Using the probability distribution table, what is the value of P(X = 2 or X = 0)? X 0 1 2 3 4 5 P 0.3 0.05 0.1 0.15 0.15 0.25 P(X = 2 or X = 0) = _____ (Points : 1) ...
1
vote
1answer
26 views

How to determine long-run probability using conditional probability?

How to determine long-run probability on a calculator and manually? For example: Ben plays a tennis match every day. If he wins on one particular day, the probability that he wins the next day is ...
-1
votes
2answers
37 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 $\frac{k}{1-0.8}=1$ Hence, $P(Z=z) = 0.2(0.8)^x$ But How do we determine the mean or variance with unrestricted z ...
0
votes
1answer
17 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
31 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
22 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 ...
3
votes
2answers
81 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} ...
3
votes
2answers
50 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 ...
0
votes
0answers
28 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 ...
1
vote
1answer
45 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 ...
4
votes
1answer
55 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 ...
0
votes
0answers
24 views

Testing for the independence of random variables

In probability theory, $X$ and $Y$ are independent if: $f_{X|Y}(x|y)=f_X(x)f_Y(y)$ If I have sample $Y_1,...,Y_n$ and I would like to test if $Y_i$ is independent from the rest of the sample, I ...
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 ...
0
votes
1answer
24 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
30 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
32 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
25 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
21 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
37 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 ...
1
vote
2answers
33 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
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 ...
-1
votes
0answers
46 views

Vector distribution after Girsanov transform

Let $X$ be a gaussian vector under $P$ and $U$ a variable such that the vector $(X,U)$ is gaussian. $dQ = Y dP$ with $Y = e^{(U −E_p(U) − 1/2 var_p[U])}$. I have to show that $X$ is gaussian under ...
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
30 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 ...
0
votes
1answer
28 views

Question on chi square transformation

I'm trying to follow the maths of a population genetics paper. Basically, I am stuck in a manipulation where the authors notice that a given function has the shape of a chi square distribution and ...
0
votes
1answer
14 views

Find the probability generating function of a GW process

Consider a Galton-Watson process with offspring distribution $\mathrm{Poisson}(1)$. That is, $\textbf{p}(k) = \frac{e^{-1}}{k!}$. Given this information, and that $P(z) = ...
1
vote
1answer
41 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
2answers
44 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 ...
2
votes
0answers
42 views

Simplifying an integral involving Gaussian PDF

Let $\phi(x)$ be the standard Gaussian probability density function and $1<Y<2$.Consider the integral $$ \int_{x=0}^\infty \int_{y=0}^\infty ...
2
votes
2answers
44 views

How to find out number of possible outcomes by trying over and over?

While working on my network exploration tool project, I've ran across the problem of reliably determining number of possible exit addresses of a tunnel with single entrance. I've came up with ...