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

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4
votes
1answer
46 views

What are general sets of conditions in which the inequality, $E\left(\frac{X}{Y}\right) \geq 1$ holds for positive r.v.'s $X$ and $Y$?

I am currently trying to determine what sorts of general conditions will allow the inequality, $E\left(\frac{X}{Y}\right) \geq 1$ to hold for positive r.v.'s $X$ and $Y$. One condition is to let $X$ ...
0
votes
1answer
48 views

Showing existence of limit of a sequence

Consider the real-valued random variable $X$ defined on the probability space $(\Omega, \mathcal{A}, P)$. Consider a sequence of positive real numbers $\{\epsilon_n\}_n$ such that $\lim_{n \rightarrow ...
2
votes
1answer
29 views

Find a function of the independent random variables X, Y, and U that has the moment generating function $[M_{X}(t) +M_{Y}(t)]/2$.

Here $X$, $Y$, and $U$ are independent random variables, $X$ and $Y$ have moment generating functions $M_{X}(t)$ and $M_{Y}(t)$ respectively, and $U$ has the uniform distribution on $[0,1]$. I can't ...
2
votes
1answer
28 views

Show that the moment generating function of $ W$ is $M_W(t) = (qe^t+p)^n$

If $Y$ is a random variable with moment-generating function $M_Y(t)$ and if $W$ is given by $W=aY+b$, then the moment generating function of $W$ is $e^{tb}M_Y(at)$ Suppose that $Y$ is a binomial ...
2
votes
1answer
28 views

how to find the expected value of a joint density function?

I have to find the covariance of this joint density function $$f(x,y) = \begin{cases} \ xe^{-x(y+1)}, & \text{if } x > 0,\ y > 0 \\[2ex] 0, & \text{otherwise} \end{cases}$$ for $E[XY]$ ...
1
vote
0answers
15 views

What is Hoeffding's inequality in Hilbert space?

Suppose I have random variables $X_1, X_2,...,X_n \in \mathcal{H}$, where $ \mathcal{H}$ is some Hilbert space. How can I bound the following term - $ P(\| \sum_{i = 1}^n X_i - E[X_i] ...
1
vote
1answer
80 views

what bounds do I integrate over to find the $cdf$ of $Z=XY$ using the following joint density function

what bounds do I integrate over to find the $cdf$ of $Z=XY$ using the following joint density function $$f(x,y) = \begin{cases} \ xe^{-x(y+1)}, & \text{if}\quad x \gt 0, y \gt 0 \\[2ex] 0, & ...
0
votes
0answers
15 views

Statistical testing, and example about critical area…

So I do not understand one part of the answer to the following question, here it goes from the beginning : A coin is thrown $100$ times and it is counted the number of times heads and tails come up. ...
1
vote
3answers
32 views

P.D.F of a mapped function of a uniformly distributed random variable

I have a random variable U which is uniformly distributed over [0,1]; Now $$X=-2logU$$ Then what would be the P.D.F. of X? I know that P.D.F of U is 1 for [0,1] and 0 otherwise so the limits of X ...
0
votes
1answer
20 views

A question with a continuous limit to a series of discrete random variables

I am trying to answer the following question- Let $X_n\sim Geom(\frac\lambda n)$ and $T_n=\frac{X_n}n$. Show that for $n\rightarrow\infty$ the distribution function of $T_n$ converges to the DF of ...
0
votes
0answers
15 views

Quantized random vector

Consider system to be: $\bar y= W\bar h+ \bar v$ where, $\bar v$ is Gaussian random vector with mean zero and covariance $R_{v}$ So, from $\bar y $ and the estimated vector $\hat y$, error is ...
2
votes
2answers
35 views

$X_k$ are $\mathcal U(0,2\theta)$ distributed, and $Y_n=\max_{1\leq k < n}X_k$, how is $F_{Y_n}=(\frac{t-\theta}{\theta})$

$X_k$ are $\mathcal U(0,2\theta)$ distributed, and $Y_n=\max_{1\leq k < n}X_k$, how is $F_{Y_n}(t)=(\frac{t-\theta}{\theta}),\theta<t<2\theta\ \ ?$ $F_{Y_n}(t) \text{ aka }(CDF)$; This does ...
2
votes
2answers
37 views

Finding Expectation and Variance of $X_1$ and $X_2$

Let $X_1$ and $X_2$ be random variables such that $E(X_i)=μ_i$ and $Var(X_i)=α_i^2$. A. Find $E(X_1+X_2)$ and $E(X_1-X_2)$ in terms of the μ's and α's. B. Suppose that $E(X_1X_2)=α$. Find ...
2
votes
3answers
64 views

Expectation of the number of times a coin is thrown until the appearance of a second “tail”

$X$ is the random variable that signifies the number of times a coin is thrown such until the appearance of a second "tail". With the probability of an appearance of "tails" on one toss being $p$. ...
1
vote
1answer
27 views

$X$ is exponentially distributed $\varepsilon(\theta)$. Using the Method of Maximum likelihood find the best (marking?)Question its centeredness

$X$ is exponentially distributed $\varepsilon(\theta)$. Using the Method of Maximum likelihood find the best (marking?)of sample $n$ for parameter $\theta$ .Question its centeredness and existence. ...
3
votes
0answers
25 views

A sequence converging to 0 in probability times a sequence bounded in probability

I'm trying to prove the following from Lehman's "Elements of Large Sample Theory" Lemma 2.3.1: If the sequence $\{Y_n, n=1,2,\ldots\}$ is bounded in probability and if $\{C_n\}$ is a sequence of ...
2
votes
1answer
32 views

Show that law of large numbers hold

Given,Xi's are iid random variables and $$f(x)= \frac{1+δ}{x^{2+δ}}$$ $δ>0$ and $X>1$ To show that law of larger numbers hold, I used khinchin's theorem which states that if Xi's are iid then ...
0
votes
1answer
24 views

Conditional Expectation w.r.t discrete Random variable

I am sorry that this is a rather trivial question but would very much appreciate a short answer. Let $(\Omega,P(\Omega),\mathbb P)$ be a probability space where $\Omega=\{1,2,3,4,5,6\}$ and $\mathbb ...
2
votes
1answer
34 views

Conditional expectation, specific function, three intervals

Let $\Omega= [0,1]$, $P$ be Lebesgue measure. Let $$Y(x) = \begin{cases} x^2, & \mbox{if } x \in [0, \frac{1}{3}) \\ \frac{1}{9}, & \mbox{if } x \in [\frac{1}{3}, \frac{2}{3}) \\ (x-1)^2, ...
0
votes
1answer
15 views

Find the moment generating function of the random variable

$f_Y(y)=e^{-3y}+\frac{2}{3}e^{-y}$ if $y>0$ and $0$ else a)find the MGF $M_Y(t)$ of Y (be careful to declare the domain) The MGF is $E[e^{tY}]$ but is that $\int_{0}^{\infty} e^{ty}f(y) dy$ because ...
1
vote
1answer
25 views

Probability: Deriving The Moment Generating Function Given the Definition of a Continuous Random Variable

Here's a question I'm working on in my textbook. It says: Let $Y$ be a Normally distributed random variable with mean $μ$ and variance $σ^2$ . Derive the moment-generating function of $X = ...
2
votes
1answer
19 views

Stats: Confidence Interval and Upper Limit

A random sample of n = 18 E-glass fibre test specimens of a certain type yielded a sample average interfacial heard yield of 40 and a standard deviation of 4. Assume the interfacial shear yield ...
0
votes
2answers
30 views

simple probability equation dilemma

Let's say I have two dependent random variables $X,Y$ and I have the following conditions: if $ Y \gt a$ then $f=P(X>a)$ if $ Y \le a$ then $f=P(X>b)$ then which formula is the right: $$ ...
0
votes
1answer
49 views

Poisson Process- Earthquakes

I am new to the concept of Poisson Process and I seem to be missing something. Earthquakes occur in a given region in accordance with a Poisson process with rate 5 per year a) What is the ...
1
vote
1answer
43 views

Expectation and variance of stock value of company after n days

Stock value of company $X$, at the beginning of day 1, is $Q$. After each day, stock value of $X$ either raises by a factor of $(1+\epsilon)$ with probability $p$, or drops by a factor $(1-\epsilon)$ ...
1
vote
0answers
40 views

Problem with $n$ balls are placed into $n$ boxes using the Indicator Method

There are $n$ balls are placed into $n$ boxes, let $N_2$ be the number of boxes with exactly $2$ balls. Find the probability that $N_2 = n$ and the probability that $N_2 = n-1$. Use the method of ...
1
vote
1answer
35 views

Conditional Distribution of Poisson Random Variables

Let $X_{1}, \dots, X_{n}$ be independent Poisson random variables (parameter $\lambda$), and set $Y=\sum_{i=1}^{n}X_{i}$. Find the conditional probability mass function of the random variable $X_{i}$ ...
1
vote
2answers
33 views

just by looking at the graph of the density function can you determine if the expected value is finite or infinite?

for example can you conclude that the expected value of $Z$ is infinite just by looking at the graph? $$f_Z(z) = \begin{cases} {\dfrac{\ln(z)}{z^2}}, & \text{for $z \ge 1$} \\[2ex] 0, & ...
3
votes
3answers
43 views

Is the expected value of this density function $Z$ finite?

Is the expected value of this density function $Z$ finite? $$f_Z(z) = \begin{cases} {\dfrac{\ln(z)}{z^2}}, & \text{for $z \ge 1$} \\[2ex] 0, & \text{for $z \lt 1$} \end{cases}$$ I know ...
0
votes
0answers
36 views

What is the $\mathbb E[(X-\mu)g(X)]$

What is the $\mathbb E[(X-\mu)g(X)]$? Let $X\sim N(\mu , \sigma ^2)$ and $g:R \Rightarrow R$ and $g$ is bounded and differentiable. I need to show $E[(X-\mu)g(X)]=\sigma ^2 \mathbb E[g'(X)]$ My ...
0
votes
0answers
39 views

Function $f$ that “magnifies” the correlation between $f(X_1, \dots, X_n)$ and $f(Y_1, \dots, Y_n)$, if $X_i=Y_i$ for some $i$.

I am trying to phrase properly (and either gain insight or answer) a question on "increasing correlation" between two r.v.'s obtained by combining either two independent families of random variables, ...
1
vote
0answers
36 views

Why does $P(Z_{1}\leq x_{1},…,Z_{n}\leq x_{n},M>u) $ equal the following expression?

We consider the following setting: Let $Z_{1},...,Z_{n}$ be iid random variables with distribution function $H_{Z}$ and $u>0$ a constant. We set $M:= \sup_{n\in N} \sum_{k=1}^{n} Z_{k} $. In the ...
1
vote
1answer
41 views

Does this conditional independence hold?

I have a random variable $X$, where $0<X<1$; and a random variable $Y$. Assume $X$ and $Y$ are uncorrelated but not independent. If I let $Z \sim binomial(p=X)$. $Z=X$ with $p=X$ and $Z=1-X$ ...
2
votes
0answers
19 views

Difference of random variables conditioned on their sum

Consider $\Omega = [0,1] \times [0,1]$ with sigma algebra of borel sets on $[0,1]^2$. Let $P$ be the Lebesgue measure on $\Omega$. Let $$\xi(x, y) = x, \ \ \ \eta(x,y) = y.$$ How can I find ...
0
votes
0answers
30 views

Uniform Sampling & CDF inverse

I have a probability exam soon, and our prof told us to study the following question: "Describe a procedure for generating independent identically distributed (i.i.d.) samples of a random variable ...
3
votes
0answers
28 views

Convergence in distribution, in $L^p$ and convergence of first and second moments

For some application, we have the following three assumptions about a sequence of Random Variables $X_n$ (with values in $\mathbb{R}^+$, $n \geq 1$: There exists a $X \geq 0$ such that a) $X_n ...
0
votes
1answer
39 views

Best possible distribution for solving maximum-likelihood for a staircase data?

I have $n$ iid sample data $x_1,x_2,x_3..., x_n$ from a probability distribution function . The sample density is defined over $[0,1]$ and is of the form: $$f(x) = \left\{\matrix{a, & ...
1
vote
1answer
61 views

how to graph Z on the (x, y) plane and how to find the cdf of Z?

We consider a random point $(X, Y)$ chosen with the following joiint density $$f(x,y) = \begin{cases} \frac{1}{x^2y^2}, & \text{if } x \ge 1 \text{ and } y \ge 1 \\[2ex] 0, & \text{else} ...
0
votes
0answers
33 views

Variance Bound for a Random Variable

Let $X$ b a random variable such that $P(a \leq x \leq b)=1$ for $-\infty<a<b<\infty$. Show that: $Var(X) \leq \displaystyle\frac{(b-a)^2}{4}$ I'm not sure on how to get started with the ...
1
vote
1answer
39 views

joint probability of two normal variables

let's say I have two random variables $$ X\sim Normal(1,\frac{2}{N_1})$$ $$Y\sim Normal(1,\frac{2}{N_2} )$$ Now if $$P(X \ge\lambda_1 ) = P(Y \ge \lambda)$$ Then is it correct to say that: $$P(X \lt ...
0
votes
2answers
27 views

Calculate probability of normal distribution

Question: Suppose that a random sample of 16 observations is drawn from a normal distribution with mean μ and standard deviation 12; and that independently another random sample of 25 observations is ...
0
votes
1answer
15 views

Prove that two random variables are dependent

Given two random variables X and Y where X is uniformly distributed on [-1,1] and Y = X^2, prove that these two random variables are dependent. Of course, it's clear that they are dependent. But, how ...
0
votes
1answer
27 views

The probability that the sum of two exponential random variables is smaller than $2$

The time it takes to service a car is an exponential random variable with rate 1. If A.J.'s car and M.J.'s car are both brought in at time 0, with work starting on M.J.'s car only when A.J.'s car has ...
1
vote
1answer
30 views

Distribution function and conditional expectation

This might be a strange question and I'm completely confused. Let $(\Omega,\mathscr{F},P)$ be a probability space and $\xi:\Omega\rightarrow\mathbb{R}$ be a random variable. Let ...
1
vote
2answers
47 views

Conditional Probabilities and Independence

I have a simple question about conditional probabilities and independence, suppose that $X, Y$ are independent random variables while as well $N$ is a random variable (which both $X$ and $Y$ are ...
1
vote
1answer
24 views

Discrete Random variable, finding pmf of X

Q: A fair coin is tossed independently 5 times. Let X denote the difference between the number of heads and the number of tails obtained. find the probability mass function of X here's my take: ...
1
vote
0answers
24 views

Probability distribution of $g^h f f^h g$

We define an $k \times k$ complex matrix $M=[V \, \mathbf{0}]$, where matrix $V$ is $k \times (k-l)$ dimensional and is unitary, and $\mathbf{0}$ is the $k \times l$ zero matrix. Let vector $f$ be a ...
0
votes
1answer
11 views

Integral of dependent uniform RVs

Help please, If I have a: $X\sim U[0,a]$ s.t. $a>0$ $Y|X=x\sim U[0,x]$ then: $$f_Y (y)= \int_{-\infty}^{\infty} f_{X,Y}(x,y)dx = \int_{0}^{a} f_{Y|X}(y)\cdot f_X(x)dx=\int_{0}^{a} \frac{1}{ax} dx = ...
0
votes
0answers
16 views

How to find $R_Y(\tau)$ if $Y(t)=X(t)\text{cos}(2\pi f_0t+\theta)$ where $\theta \sim \text{unif}[-\pi,\pi]$

Suppose $X(t)$ is wide-sense stationary (WSS) process. How to find $R_Y(\tau)$ if $Y(t)=X(t)\text{cos}(2\pi f_0t+\theta)$ where $\theta \sim \text{unif}[-\pi,\pi]$. The answer is: ...
0
votes
3answers
73 views

how to find the distribution of $Z=X_1/X_2$?

If $X_1$ and $X_2$ are independent exponential random variables with respective parameters $\lambda_1$ and $\lambda_2$, find the distribution of $Z=X_1/X_2$. Also compute $P(X_1 \lt X_2)$. What ...