Questions about maps from a probability space to a measure space which are measurable.

learn more… | top users | synonyms

0
votes
0answers
18 views

Mixed distribution of product of Bernoullie and Gaussian r.v

Confused with the formulation of density function of the following mixed distributed random variable $Z$. $$Z \equiv X \cdot Y,$$ where $( \cdot)$ is product operation, and $X$ and $Y$ being ...
0
votes
0answers
23 views

A Gaussian Divided by a Gaussian Equal to A Gaussian Divided by a Constant

I have a neural-network model in which each neuron is associated with an angle $\theta$. Firing rate as a function of $\theta$ is either a Gaussian or a constant. The claim has been made using this ...
0
votes
2answers
26 views

A problem on continuous random variables

I was reading a The First course on Probability by Sheldon Ross, while I stuck at this possibly stupid doubt. The problem is : The density function of X is given by $$ f(x) = \begin{cases} 2x, ...
0
votes
1answer
31 views

Can anyone help clarifying the geometry in this probability, random variables question.

So basically the question is to find the CDF of $Z$ where $Z$ is the random variable that signifies the distance from a point in a square(sides 1 length) to a fixed vertex of the square. I do not ...
2
votes
1answer
33 views

Random variable $X$ is given with the density function $ \phi_X (x)= \frac{1}{2} e^{-|x|}$ Find the distribution of the random variable $Y$ if:

$$Y=\begin{cases}-X-2,\ \ \ \ X \leq -1 \\ \ \ \ X, \ \ \ \ \ -1 \leq X \leq 1 \\ \ \ \ \ 1, \ \ \ \ \ \ \ \ \ X >1 \end{cases}$$ Now I'm only interested in $t >1.$ (That is only ...
1
vote
1answer
11 views

Continuous random variables and probability density function

OK, I know that a random variable $X$ from some probability space to $\mathbb R$, with some additional properties. It is discrete if it's image in $\mathbb R$ is dicrete. It is otherwise called ...
0
votes
0answers
9 views

Random data generation from a bivariate distribution

Let $X$ and $Y$ be non negative random variables with joint distribution \begin{equation} F(x,y)=1-e^{-x}-e^{-y}+e^{-x-y-\delta xy}; ~~~x\geq 0,~~y\geq 0. \end{equation} How to generate a bivariate ...
1
vote
2answers
32 views

Proving the $Pr(d>0|a+d=\pi)$ is increasing in $\pi$ when a and d are two independent normal distributions.

I was wondering if it is possible to prove the following (or show false otherwise). Given two independently distributed random variables $a\sim \mathcal{N}(\alpha,\sigma_\alpha^2)$ $d\sim ...
2
votes
1answer
17 views

Conditional probability with max(X, Y)

Let $Y_n=$ the outcome of the $n$-th die roll, let $X_{n+1} = \max \{X_n, Y_{n+1}\}$ with $X_1=Y_1$. What is $P(X_{n+1}=j \ | X_1=i_1, ..., X_n=i)$? I know that it is $P(\max \{X_n, Y_{n+1} \}=j \ | ...
0
votes
1answer
38 views

Understanding the sum of random variables

I am currently learning probability theory. I have two questions: I would like to know through an example what is meant by the sum of random variables (r.v.). To make things simple let consider only ...
2
votes
0answers
26 views

If $X_n$ are i.i.d. $Uniform(0,1)$ then show that $S_n$ converges a.s. to $\infty$

Suppose $\{X_n\}$ is an i.i.d. $Uniform(0,1)$ sequence of random variables. Define $S_n=X_1+X_2+...+X_n$. Show that $S_n\to\infty$ almost surely. I believe I have solved the problem and I wish ...
1
vote
1answer
22 views

Two related question, in one. Same topic: Dispersion..

$1.$ Prove: If $X_1,X_2,X_3,\ldots,X_n$ are independent random variables then: $$D\left(\sum_{i=1}^n X_i\right)=\sum_{i=1}^n D(X_i)$$ Proof: Because of independence we have: $$D(\sum_{i=1}^n ...
0
votes
1answer
21 views

A die is thrown $n$ times. $X_1$-number of times a number from $\{1,2,3\}$…

.. $X_2$ number of numbers that fell from $\{4,5\}$, $X_3$ number of $6's$ that fell. Find $$P\{ X_1=k\mid X_2=m\};0\leq m \leq n.$$ Now, I believe that $X_3$ is completely irrelevant here. What I ...
0
votes
2answers
23 views

In a box which has balls numbered 1..100 , 5 balls are drawn.

$X$- random variable that represents the largest number of the 5 drawn. Find the distribution of $X$. Now, it seems that this random variable is of discrete type. What I have trouble it defining it ...
2
votes
1answer
24 views

Confusion about the sample mean and random variables

As I understand the sample mean you just add a bunch of random variables that constitute a sample from their common distribution and divide by the number of those same random variables. When I apply ...
1
vote
2answers
34 views

Basic measure theory question about $\sigma$-algebra

Let $Y, Z$ be random variables and $G$ be a $\sigma$-algebra. Page 69 of Shreve's Stochastic Calculus for Finance II says "because both $Y$ & $Z$ are $G$-measurable, their difference $Y-Z$ is as ...
0
votes
1answer
35 views

We write down the date of each person's birthday we meet (say Feb 29. doesn't exist).

Random Variable $X$ is the number on persons we met til we wrote down every date in a year. Find the expected value of $X$. Find $E(X)$- expected value. From this example I can definitely understand ...
1
vote
0answers
19 views

Showing that a sequence of random variables has CLP.

This is an exercise that I am stuck at. I managed to solve (i) and (ii), which are relatively easy. Here the Feller condition in (i) is as below. Also, the central limit theorem I learned is like ...
1
vote
1answer
18 views

Show that $X_n/n$ does not converge almost surely

I am generally able to prove that a sequence of random variables $X_n$ converge almost surely to a random variable $X$ by using the following strategy: Take any typical sample point ...
3
votes
3answers
32 views

Property of cumulative distribution function

I was taking the course on random variables , where I faced below property of cumulative distribution function: $$\lim_{x\rightarrow a^+}F_X(x)=F_X(a^+)=F_X(a)\qquad\qquad ...
1
vote
1answer
29 views

Coupon Collector's Problem — Expected Value of each item

So I guess my problem is based on the famous coupon collector's problem, which is, if you should not be familiar with it, the following: Given N different coupons from which coupons are being drawn ...
1
vote
0answers
68 views
+50

Probability of absorption of a biased random walk when the starting point has binomial distribution

Consider a random walk $\{0,1, ... , N\}$ with up probability $p$ and down probability of $p-1$ where $p \neq 1/2$. Suppose there are absorbing barriers at $0$ and $N$ and that the starting point, ...
1
vote
0answers
18 views

What is the pdf of $X$, where $dX_t = -aX_t + d N_t, N_t$ is a compound Poisson process?

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where (I'm not sure this notation makes sense, I'm not very familiar with the stochastic calculus ...
0
votes
1answer
53 views

Distribution of Bernoulli and Uniform Random Variable

Here's a problem I am stuck on: Let $X$ and $Y$ be independent random variables such that $X$ is Bernoulli-distributed with $p=1/2$, and $Y$ is uniformly distributed on the interval $[0,1]$. Then: ...
0
votes
1answer
26 views

What is the probability that on a given day, the number of half gallon containers provided is enough?

In a grocery store 400 customers shop every day. The number of half gallons of nonfat milk bought by a randomly selected customer is a random variable X having P(X=0)=0.3, P(X=1)=0.5, and P(X=2)=0.2. ...
0
votes
2answers
33 views

Normal Distribution of Sums

I have two normally distributed random variables $X$ and $Y$. Then I know that the sum $X-Y$ is also normally distributed (i). However, I want to show (preferably by a counter example) that the ...
1
vote
0answers
10 views

On random functions taking values in the space of continous functions.

I upload the image of the passage that is unclear to me (Theoretical statistics by Keener): I do not understand how $W_1, W_2, \dots$ take values in $C(K)$. For every different realization of the ...
2
votes
1answer
29 views

How to represent $Prob(X_1+X_2 \leq a, X_2+X_3 \leq b, X_3 +X_4 > c)$ with mutually independent random variables?

There are four mutually independent random variables: $$X_i : \Omega \to \mathbb R$$ for $i= 1,2,3,4$ The cumulative distribution function of them is given as $F_i(x_i)$. How to represent ...
-2
votes
0answers
13 views

Expected value of exponential random variable [closed]

If an exponential random variable, X, has failure rate λ, what is E[X|X<λ]? I'm not sure how to start here. I know that E[X] = 1 / λ for an exponential random variable. Is the probability that X ...
0
votes
0answers
17 views

Expected value of exponential random variable

If an exponential random variable, X, has failure rate λ, what is E[X|X<λ]? I'm not sure how to start here. I know that E[X] = 1 / λ for an exponential random variable. Is the probability that X ...
0
votes
1answer
20 views

Expected Value: how to understand this expression?

So I have come across a question asked by my peers. Define: $$g:=\sqrt{E[|y_r(t)|^2]}$$ Given that $$y_r(t)=\sqrt{t}\cdot h+b+k+c,$$ where $h$, $b$, $k$, and $c$ are independent random variables. ...
0
votes
0answers
13 views

Posterior probability estimation in MAP model

I have a question about probability. I am using Bayes rule to determine which class the $x$ belong to. According to Bayesian formula, the MAP estimation is equivalently found by $$p(x \in \Omega_i|x)= ...
0
votes
3answers
26 views

Given the joint density function for X~Unf(0,2) & Y~Unf(0,3) find Pr(XY < 1)

I have two independent random variables, X~Unf(0,2) & Y~Unf(0,3). Their joint density function is f(x,y) = 1/6 if 0<=x<=2 and 0<=y<=3 else f(x,y) = 0. I'm supposed to find Pr(XY < ...
7
votes
1answer
99 views
+100

Mean value theorem for random variables (inside an expectation value)

In a proof I am trying to understand a mean value theorem for random variables is used. It is stated that $$E[f(X+Y)]=E[f(X)+E[f^\prime(X+\theta Y)]Y]$$ for real valued random variables $X$ and $Y$ ...
0
votes
1answer
34 views

What is the Difference in the Average and the Mathematical Expectation in the following Problem

Suppose that a school has 20 classes: 16 with 25 students in each, three with 100 students in each, and one with 300 students, for a total of 1000 students. The average class size is simply ...
0
votes
1answer
32 views

The probability of passing a probability course with limitations. [closed]

John is attempting to pass a probability exam again and again until he has succeded. However, he is allowed to try only $n$ times. Suppose that the probability that John doesn't flunk is $p$ ...
0
votes
0answers
32 views

How can two random variables are continuous infers that their jointly random variable is continuous?

We assume that $\forall a,b$ such that $a2+b2>0$, $aX+bY $ is continuous random variable. But we don't assume that $X$ and $Y$ are independent. My question is the following: Under which ...
5
votes
2answers
46 views

Sequence satisfies weak law of large numbers but doesn't satisfy strong law of large numbers

Let $\{X_n\}_{n=1}^{\infty}$ be a sequence of independent random variables such that $$P(X_n=n+1)=P(X_n=-(n+1))=\frac{1}{2(n+1)\log(n+1)}$$ $$P(X_n=0)=1-\frac{1}{(n+1)\log(n+1)}$$ Prove that $X_n$ ...
1
vote
1answer
14 views

Probability of event in join sample space of X & Y

From what I understand, the answer should be $(0.1+0.35+0.05)$, since the given points have probability summation $1$. Am I correct? By the way, the correct answer unknown.
0
votes
0answers
19 views

How to compute the average power of an ergodic process?

Rxx(0)=3 is the average power and if i take limit as t goes to infinity i will get the (E[x])^2 to get variance you subtract 3-2 = 1 is this correct ? and can someone tell the difference ...
0
votes
1answer
5 views

Variance function is variance stabilising

Y has mean u and variance function V(u). If $V(u) = \alpha.u^v$ then $h(y) = y^{(2-v)/2}$ is variance stabilising which means that Var(h(Y)) is approximately constant. I tried to prove it computing ...
0
votes
1answer
17 views

Finding an interval estimate for $\mu$ given a sample size and variance

I'm in a statistics class and am doing a problem for homework about confidence intervals. I don't really know what it's asking though or when I've even reached a valid solution. The problem says: ...
2
votes
2answers
27 views

Expectation over multiple variable?

So I have come across a question asked by my peers. Define: $$g:=\sqrt{E[|y_r(t)|^2]}$$ Given that $$y_r(t)=\sqrt{t}\cdot h+k,$$ where $h$ and $k$ are independent random variables with variance ...
1
vote
1answer
24 views

How to calculate the probility of 2 independent events of having same value?

We are learning to calculate the probability of sums and difference of random numbers. Here is the problem: One athlete knows from past experience that the distances of his javelin throws follow a ...
1
vote
2answers
32 views

Uniform distributed variable X:U(-9,9) is given. Find the CDF of Y if Y is..

$$Y= \begin{cases} 4X,\ \ \ |X| \leq 3 \\ 0,\ \ \ \ \ \ |X|>3 \end{cases}$$ My take on it: $$F_Y(y)=0; y\leq-12;F_Y(y)=1; y\geq 12;$$ $$F_Y(y)=\{Y < y \}$$ In class in a similar task we ...
0
votes
1answer
41 views

Proof conditional probability formula

a question for my homework for probability goes as follows: Given X,Y,U, three discrete random variables, prove the following: $$ p_{X|Y}(i|j) = \sum_{k=0}^{\infty}p_{X|YU}(i|j,k)p_{U|Y}(k|j) $$ The ...
1
vote
2answers
25 views

Probability distribution of number of columns that has two even numbers in a chart

We distribute numbers $\{1,2,...,10\}$ in random to the following chart: Let $X$ be the number of columns that has two even numbers. What is the distribution of $X$? My attempt: ...
0
votes
1answer
52 views

Random Variable $X= U:(0,4)$ is given. Find the CDF of $\min \{ |X-1|, 5-X\}+1$

Random Variable $X= U:(0,4)$ is given. Find the CDF of $Y=\min \{ |X-1|, 5-X\}+1$ X has uniform distribution. So we know that $$Y\in (1,6)$$, therefore $$y\leq 1 \ \ \ F_y(y)=0 ; y\geq6\ \ \ ...
0
votes
0answers
7 views

specific examples of random variables satisfying a given condition.

Theorems such as the central limit theorem only says random variables satisfying certain conditions have some properties. Now, what I am curious about is the existence of such random variables. For ...
1
vote
1answer
33 views

Expected value of doubling or halving a number with equal probability

I have this question that you start with a value say c. At each step, you either double or half the value with equal probability. Let $X_i$ be the value of c at ith-step, I need to find the expected ...