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

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0
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1answer
42 views

Characteristic function and cdf

How do we find the points of discontinuity of a distribution of a random variable if the characteristic function $\phi (t)$ is given? How to find the cdf of a random variable X if the characteristic ...
2
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0answers
44 views

When to stop pumping up balloons?

Yesterday I acted as a volunteer in a psychology/neurology experiment where one of the trials consisted of playing a computer game in which you had to click the mouse to pump up a balloon. For each ...
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2answers
23 views

Determine individual distribution when knowing joint distribution of $2$ random variables X and Y

I have the joint distribution of $2$ random variables $X$ and $Y$. Here: How can I determine the distribution of $X$ and the distribution of $Y$ knowing this? I have tried using the following ...
5
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2answers
73 views

Probability: mathematically what does it mean to say “let $X$ be a random variable WITH a cdf/pdf”

I don't quite understand what people mean by let "$X$ be a random variable WITH a cdf/pdf". For example, there is a question that says: "Let X be a random variable with the 3-parameter Weibull pdf and ...
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2answers
27 views

what is the distribution of $\sum_{i=1}^n a_i^2$, where $a_i$ is independent Gaussian variable with zero mean and $b_i$ variance

If all the variance of all the Gaussian variables are the same, saying 1, then the distribution is Chi-squared https://en.wikipedia.org/wiki/Chi-squared_distribution. In my question, the variance are ...
0
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0answers
24 views

Gamma distribution of $Y^2 \sim Γ(0.5,0.5)$

So the question asks: Let $X\sim Γ (s,λ )$ be a random variable distributed according to a gamma distribution (with $s$, $λ > 0$). Suppose $Y$ is a standard normal random variable. Show that $Y^2 ...
0
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1answer
35 views

What are the expected value and the standard deviation of the number of games the final will take?

In the final of the World Series Baseball, two teams play a series consisting of at most seven games until one of the two teams has won four games. Two unevenly matched teams are pitted against each ...
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1answer
24 views

Conditional joint density of $Z=Y/X$

So the question asks : Let X and Y be random variable with joint density: $f_{X,Y}(x,y) = c $ when $x^2 +y^2 ≤1 $ $f_{X,Y}(x,y) = 0 $ in other situations (a) Find the value of the constant $c$. ...
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0answers
26 views

Covariance of functions of Gaussian random Variables

The two random variables $x$ and $y$ are Gaussian distributed with some Covariance. f is a non-linear stationary function. Now the following algorithm is applied: $ \underline{Algorithm:} ...
0
votes
1answer
26 views

Which is correct formula for Plot/Gibbs distribution?

Helllo all, I am using Bayesian rule to classification the data. It has two term: likelihood and prior terms. The prior term can estimate by Gibbs distribution. According to the MRF-GRF (page 12),book ...
1
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1answer
19 views

What is the distribution function of a random outcome in closed interval [0,1]

a point is thrown at random on the interval [0,1], and if the outcome is x, you get 100x dollars. Y represents the amount of money you get. What is the pdf of X? what is the pdf of Y? I thought the ...
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0answers
18 views

Preserving stationarity of a process

This question is just for personal understanding. Is there common knowledge out there of functions that preserve stationarity of a process? More concretely, suppose $x(t)$ is WSS (wide-sense ...
1
vote
1answer
43 views

Chebyshev inequality for some random variable [closed]

I am looking for some help with the chebyshev problem that I am not quite sure how to do and am stuck on. Could someone provide some advice for the following question? Let $Z_{1}$, $Z_{2}$, $Z_{3}$, ...
0
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2answers
42 views

Derivative of a random number

I would like to numerically solve a differential equation which contains a derivative of a random number (using a finite difference method with a time step $\Delta t$). Let say I need to solve for ...
0
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0answers
27 views

Linear approximation of a random vector (X,Y) in the form of Y = aX + b

Problem I have a random variable vector (X,Y) whose joint mass function is given by the below table (X = j, Y = k). Additionally, the correlation of the vector is given by : E[XY] = probability of ...
0
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1answer
29 views

What does $X\sim Pois(\lambda S)$ mean?

I'm wondering what exactly does $X\sim Pois(\lambda S)$ mean when $S$ is a random variable as well? I guess $\mathbb{P}(X=k)=\frac{(\lambda S)^k}{k!}e^{-\lambda S}$ but still I do not know what that ...
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0answers
12 views

Var(X) using the exponential distribution family for the Gamma distribution

I have written the Gamma distribution ($\frac{1}{\Gamma(\mu)}r^\mu x^{\mu - 1}e^{-rx}$) in the normal form for the exponential family of distributions which is given as follows; $$exp\{\eta_1 ...
0
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1answer
53 views

Conditional entropy under quantization

Let $X$ be a continuous random variable and $X^n$ its quantization that becomes finer with larger $n$. Let $Y$ be a deterministic function of $X$. Then we have that the conditional entropy $$H(Y|X) = ...
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0answers
26 views

Independent composed random variables

Let me ask if it is possible to prove the following. Let $X_i, \, i=1,\ldots,n$ iid random variables and $Y_i, \, i=1,\ldots,n$ iid random variables. Now we define the composed random variables $$ U = ...
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0answers
19 views

Probability of bike rack being filled given distributions of bikes leaving and returning

Say we have $m$ spaces in a bike rack, which can be accessed by $n>m$ specific bikes. A bike leaves the rack at a random time $t$ according to a distribution $P_l(t)$ and returns a time $t'$ ...
1
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1answer
28 views

If $X \sim \text{Exp}(1)$, show that $\sqrt X$ follows a Rayleigh distribution.

I have a task to show that if $$X \sim \text{Exp}(1)$$ then $$\sqrt X \sim \mathrm{Rayleigh}\left(\frac{1}{\sqrt 2}\right)$$ How do I start from here? For $X\sim$ exponential, I know that the pdf ...
2
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4answers
64 views

What do I take as the random variable?

There are 500 misprints in a book of 500 pages. What is the probability that a given page will contain at most 3 misprints? How do I solve this? What do I take as the random variable here?
1
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1answer
32 views

How do we find the mean and variance of a single random variable given two other variables?

Let $Y_1 =\frac{1}{2}X^2 − 1$ and $Y_2 =\frac{1}{2}X − 1$, where $X$ is a random variable whose mean is positive. Moreover, we know that the mean of $Y_1$ is $2$, and the variance of $Y_2$ is ...
0
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0answers
28 views

Expected value of a random variable transformed by exponential

I need to compute $$E[e^{-bx}]$$ where $x$ is drawn from an exponential with rate $a$ truncated to interval $[0,c]$. Is the following approach correct? $$E[e^{-bX}] = \frac{\int_0^c{e^{-bx}\cdot a ...
2
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1answer
42 views

Is it possible that $X>\operatorname E[X]$ for a random variable $X$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space and $X\in\mathcal L^1(\operatorname P)$. I'm wondering whether or not we can prove that $\exists\omega\in\Omega$ with ...
0
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2answers
20 views

Choosing the variance of a random normal variable to fulfill some criteria

Suppose you create a random normal variable $X$ with mean zero and variance $\sigma^2$. You wish to choose $\sigma^2$ such that 80% of the time (or $a$% of the time, to make it more general), $X$ is ...
0
votes
1answer
47 views

Square of a uniform random variable

Let $X$ be a random variable that follows $Uniform(-1,2)$ distribution. Then what will be the cdf of $X^2$? What i did was $$P(X^2\le y)=P(X\le y^{0.5})= ...
1
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1answer
29 views

find joint and marginal density functions [duplicate]

I'm trying to figure out the solution to the following: Given: a) $Z_1$ and $Z_2$ are independent standard normal random variables b) $Y_1=Z_1$ and $Y_2 = Z_2\sqrt{1-\rho^2} + \rho Z_1$ Find: ...
3
votes
2answers
38 views

How to show that $y=Px$ is distributed like binary $x$ for random permutation $P$?

Drawing a random binary vector $X\in\{0,1\}^n$ from the uniform distribution, the probability $\mathbb{P}(X=x)$ to get a specific $x\in\{0,1\}^n$ is known ($=\frac{1}{2^n}$). Let ...
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0answers
14 views

Pearson correlation of neural responses with it's linear estimation

I am trying to anderstand the following fact: Suppose I have a linear estimation of a stimulus: $ \hat{s} = \mathbf{w}^T(\mathbf{r} - \mathbf{f}(s_0)) + s_0$ where $\mathbf{w}$ is a vector of ...
0
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0answers
51 views

Probability at a point

If X is a continuous random variable then we say that probability at a point is neglected or id zero.but if it is discrete then the probablity of that point will be taken into consideration I know i ...
7
votes
1answer
95 views

Gambling system theorem given by Doob

Let$\{X_n\}_{n=1}^{\infty}$ be a sequence of i.i.d. random variable. Let $\{\alpha_k\}_{k=1}^{\infty} $be a sequence of strictly increasing finite stopping times. Then ...
3
votes
2answers
85 views

Probability: Store opening time

Smith has a small booth where he sells lottery tickets. Customers arrive according to a Poisson process of rate $\lambda$= 1 per minute. He will close the shop on the 1st occasion that $a$ minutes ...
5
votes
1answer
63 views

Crossing a lane of traffic

A pedestrian wishes to cross a single lane of fast-moving tra c. Suppose the number of vehicles that have passed by time t is a Poisson process of rate , and suppose it takes time a to walk across the ...
1
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1answer
22 views

Showing $X_n\rightarrow_P X$ and $X_n\rightarrow_P Y$ $\implies$ $P(X=Y)=1$

A sequence of random variables $\left\{ X_n \right\}$ $\textbf{converges in probability}$ to a random variable $X$, denoted by $X_n\rightarrow_P X$, if for every $\epsilon>0$, $$ P(|X-X_n|\geq ...
0
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0answers
29 views

Brownian motion - independence

I have not so difficult task - For Brownian motion $W(s)-W(t)$ is independant of $\sigma$-algebra $F(t)$ $0\leq t<s$. My goal is to show that for $0\leq t<s<u$, $W(u)-W(s)$ is also ...
0
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0answers
63 views

Difference of two random variables

$X\,$ follows exponential $1$ and $Y$ follws Gamma distribution with mean $2$ and variance $2$.Given that $X$ and $Y$ are independent What is $P(X\lt Y)=??$ What i did was first i found the ...
1
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2answers
41 views

We flip a coin until a tail or five heads in a row occur. What is the number of expected flips?

We flip a coin until a taild or five heads in a row occur. What is the number of expected flips? I have tried to solve this by first defining 2 random variables: X ...
0
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1answer
35 views

Random variable probability

On probability space $\Omega$ with elements a,b,c,d,e. Define $\sigma$-algebra $F$ on $\Omega$-collection of subsets of $\Omega$ and $H=X+Y$. Probability measure by P{a}=P{b}=P{c}=P{d}=1/5,RV X,Y: ...
1
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1answer
65 views

Show Y has a uniform distribution if Y=F(X) where F(x)=P[X $\le$ x] is continuous in x.

If $ F(x) = P[X\le x] $ is continuous in x, show that $ Y=F(X) $ is measurable and that $Y$ has a uniform distribution $ P[Y\le y] = y, \; 0\le y \le 1 $ My first question is about notation. What ...
0
votes
1answer
37 views

Using Inequalities (Markov + Chebyshev) for lower bounds

I have an exam in a few hours and realized there's problems on practice problems that we didn't directly have to do for class. I know that the markov bound is: $P(X >= k) <= E[X] / k$ and the ...
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2answers
33 views

Convolution formula proof- Random discrete varaiables [closed]

Let X, Y be discrete random variables and take values at $1, 2, · · · , n, · · · $ $f_{X}(t)=\sum_{k=0}^{k=inf} P(X=k)x^{k}$ is the probability generating function. and this result was given below ...
2
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0answers
34 views

Ratio of Order Statistics goes to infinity?

Is there a sequence of distributions $(F_n)$ with $F_n(0)=0$ such that $\frac{E[\max(X_{1,n},X_{2,n})]}{E[X_{1,n}]} \to +\infty $ as $n\to +\infty$, where $X_{1,n}$ and $X_{2,n}$ are ...
1
vote
1answer
26 views

Conditional exp. - $\mathbb{E}[Y|X]$=$X$

I have very common task - Consider a probability space $\Omega$ with four elements a,b,c,d. Define $\sigma$-algebra $F$ on $\Omega$-collection of subsets of $\Omega$. Probability measure by ...
3
votes
2answers
146 views

A DE Shaw Interview Question about Two i.i.d random variables inequality [closed]

If $X$ and $Y$ are i.i.d positive random variables, Prove that $\Bbb E(X/Y) \ge 1$: I use Jensen's inequality $\Bbb E[\exp(\log(X/Y))]$ and get the answer. One can also use the A-G inequality to ...
1
vote
2answers
30 views

Moment Generating Function of the Product of Three I.I.D Bernoulli random variables

Let X1, X2, X3 be i.i.d. random variables with distribution $P(X_1 =0)=\frac13, P(X_1 =1)= \frac {2}{3}$ Calculate the moment generating function of $Y = X_1X_2X_3.$ My work: $M_x(t)= E(e^{tx}) = ...
2
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1answer
21 views

Can the expected value of a random variable be a function of other random variables?

Let $Y$, $Z$, and $L$ be random variables and the possible value of a uniformly distributed random variable $X$ can be expressed by the following inequality $$Y+Z-L \le X \le Y+Z$$ Can we say that ...
0
votes
1answer
33 views

Density of the Sum of Two Exponential Random Variable

For independent random variables X ∼ Exp(1) and Y ∼ Exp(2), find the density of the random variable Z = X + Y . My work: For any exponential distribution with parameter $\lambda$ the function is ...
1
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0answers
53 views

Conditional expectation $\mathbb{E}[Y|X]$

I have very common task - Consider a probability space $\Omega$ with four elements a,b,c,d. Define $\sigma$-algebra $F$ on $\Omega$-collection of subsets of $\Omega$. enter image description here ...
1
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1answer
17 views

Finding a distribution function for $Y=ax^2$

X is a discrete random variable with a distribution function $F_x$ and PMF $P_X$. How can I find the distribution function of $Y = ax^2$, a $\neq$ 0.