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

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0
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2answers
51 views

X and Y are independent random variables and their distributions are..

X and Y are independent random variables and their distributions are.. $P(X=1) = 0.1 $ $P(X=2) = 0.2$ $P(X=3) = 0.3 $ $P(X=4) = 0.4 $ $P(Y=4) = 0.4 $ $P(Y=2) = 0.3$ $P(Y=3) = 0.2 $ $P(Y=4) = 0.1$ I ...
2
votes
0answers
21 views

Density of the $k^{th}$ smallest of $X_1,X_2,…,X_n$

Show that if $(X_1,X_2,...,X_n)$ are i.i.d. with common density $f$ and distribution function $F$, then $X_{(k)}$ has density $$f_{(k)}=k\binom{n}{k}f(y)(1-F(y))^{n-k}F(y)^{k-1}$$ where ...
0
votes
1answer
26 views

Evaluating an expectation of the supremum of collection of random variables

I know that $\mathbb{E}(sup_n|X_n|)=\infty$ if $X_n=\frac{2^n}n\cdot\mathbf 1_{(1/2^{n+1},1/2^n)}$. However I am not sure how this can be evaluated explicitly. The probability space is $Ω=[0,1]$ ...
0
votes
1answer
39 views

Compute a conditional probability of normal random variable

Suppose $X, T$ are continuous random variables, and $X \sim \mathcal{N}(0, 1)$, $T$ have density function $f_T$. (But $X,T$ do not have joint density) Is there any way to compute the following ...
0
votes
1answer
26 views

$X $ and $Y$ are continuous $RVs$, such that$ f(x,y) = 2, 0\leq x\leq 1, 0\leq y\leq 1, 0\leq x+y\leq 1$

X and Y are continuous RVs, such that $f(x,y) = 2, 0\leq x\leq 1, 0\leq y\leq 1, 0\leq x+y\leq 1$ I'm trying to find $P(x<1/2,y>1/2)$. So i'm integrating from $\dfrac{1}{2}$ to $1$ for $y$ ...
0
votes
1answer
18 views

Covariance of random variables with identical distribution.

Let $X_1,...,X_n$ be random variables with identical distribution, and for all $i=1,...,n$ $\mathrm{Var}(X_i)$ exist. 1. Show that the covariance between each two random variables exist. 2. Show that ...
1
vote
1answer
51 views

First hitting time expectation and Markov property

Let $H_A$ be the first hitting time, such that $H_A\geqslant1$, so we have $X_0=i\notin A$. All texts I looked at, state without any further justification that $$ \mathbb E(H_A\mid X_1=j, ...
1
vote
0answers
1k views

Problem with the expectation of a maximum of independent gamma distributed random variables

Having a problem with the expectation of the maximum among $n$ independent random variables $ X_1, X_2 \dots X_n$ all ~ the same class of distributions but not necessarily the same mean and other ...
0
votes
1answer
47 views

Find the PDF of Y given Y=X(2-X) and X's PDF

Suppose that the continuous random variable $X$ has probability density function $f_X(x)=\begin{cases}\frac{1}{2}x & \text{if } 0<x<2\\0&\text{otherwise}\end{cases}$ Let $Y=X(2-X)$. ...
0
votes
1answer
182 views

If $X' \leq X$ almost surely, is it possible to prove that $P(X = s) \geq P(X' = s)$?

With respect to my previous question, let us define $X$ as: $$ X = \sum_j^r l^j Y^j, $$ where $l^j \geq 0$ and $Y^j$, $j = 1, \ldots, r$ is a Bernoulli random variable which takes on values in ...
3
votes
1answer
35 views

Find a sequence of r.v's satisfying the following conditions

I think part a) can be solved by using $X_n=\frac{1}{n}\chi_{[0,n^2]}$ Not sure about part b).
1
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2answers
104 views

Tossing a coin with at least $k$ consecutive heads

Toss a coin with $\Pr(\text{Heads})=p$ repeatedly. Let $A_k$ be the event that $k$ or more consecutive heads occurs amongst the tosses numbered $2^k,2^k+1,...,2^{k+1}-1$. Show that, $\Pr(A_k\ ...
0
votes
1answer
74 views

Is this true: probability independent from i?

We have a set of i.i.d. random variable $X_i$ with some discrete distribution. Further we have a random variable Y, Independent from $X_i$ with a Binomial Distribution Bin(n,p). Now we are ...
0
votes
1answer
21 views

Normal Random Variables

Let Z1 and Z2 be independent standard normal random variables. What is the probability that the minimum of Z1 and Z2 will be greater than 1.0? How do I go about this when I have no values? Is the ...
2
votes
0answers
19 views

does the hilbert space construction of random variables allow for infinite variance?

I am reading a book (Hilbert Space Methods in Probability and Statistical Inference by Small) which says that random variables can be viewed as functions in the hilbert space $L^2$ with the inner ...
0
votes
1answer
21 views

Uniform Spinner is spun twice..

A fair uniform spinner is spun twice, and the results V and W are noted. V and W are uniform RVs ∼U[0,1]. I'm trying to answer the question what is the joint pdf for V and W. I know that I have to ...
1
vote
2answers
66 views

Expected Value of Intersection of two Binomial Random Variables

Ok the problem is as follows: (I am currently studying for my first actuary exam so this isn't a specific hw question! Just trying to figure it out!) A and B will take the same 10-question exam. ...
2
votes
2answers
58 views

Find the distribution of random variable $XY+X+Y+1$

X and Y are iid with density $f(x)=\frac{1}{(1+x)^2}I_{(0,\infty)}$. Find $P(Z\le z)$ where $Z=XY+X+Y+1$ my effort: $P(Z\le z)=P((x+1)(y+1)\le z)=P(x\le ...
1
vote
1answer
33 views

If $X,Y$ are independent and geometric, then $Z=\min(X,Y)$ is also geometric

Let $X,Y$ be independent geometric random variables with parameters $\lambda$ and $\mu$. If $Z=\min(X,Y)$. Show that $Z$ is geometric and find its parameter. (Answer $\lambda\mu$) $\displaystyle ...
1
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0answers
51 views

Conditioning on function of random variable and random variable itself

Suppose that $Y_{i}\in\{0,1\}$ is a binary variable, and $X_{i}$ is some random vector in $\mathbb{R}^{d}$ . Why can we say the following: \begin{eqnarray*} ...
2
votes
1answer
21 views

Modes of Convergence of a particular random variable

Let $X_n \sim U([-1/n,1/n])$ be uniform random variables on $[-1/n,1/n]$ for $n \in \mathbb{N}$. Do the $X_n$ converge, and if yes in what sense? I think it converges pointwise as for any $x \in ...
1
vote
1answer
57 views

finding the limits of integration for joint probability

I have three variables $x_1$, $x_2$ and $x_3$. Their joint dist. is $f(x_1,x_2,x_3)= \exp(-x_1-x_3)$, where limits of $x_3 = 0$ to $\infty$, $x_2 = x_3$ to $\infty$ and $x_1 = x_2-x_3$ to $\infty$. ...
2
votes
1answer
34 views

$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), Z_2:=\sqrt{-2\log X} \sin(2\pi Y)$ independent and normal

I am looking for a nice proof of the following statement: If $X,Y\sim U(0,1)$ are two independent uniformly distributed random variables, then $$Z_1:=\sqrt{-2\log X} \cos(2\pi Y), \quad ...
2
votes
1answer
43 views

Let X be an exponential random variable with P(X < 1/3) = 0.75. What is E(X)?

Let X be an exponential random variable with P(X < 1/3) = 0.75. What is E(X)? I don't get this. Please help.
0
votes
0answers
37 views

Law of large numbers with random weights

Let $\mu_i$ be i.i.d. RVs with mean zero, and let $a_i$ be random weights that are not independent and are not identically distributed, $i=1,...,N$. $\mu_i$ is orthogonal to $a_j\;\forall j$. Is ...
2
votes
1answer
44 views

Measures in conditional expectation.

I always make confusion when a measure has to be changed in some other measure. This time I'm stuck on a change of measure in the definition of conditional expectation of a random variable. If $Z$ is ...
1
vote
3answers
46 views

Find the Mean for Non-Negative Integer-Valued Random Variable

Let $X$ be a non-negative integer-valued random variable with finite mean. Show that $$E(X)=\sum^\infty_{n=0}P(X>n)$$ This is the hint from my lecturer. "Start with the definition ...
0
votes
1answer
33 views

Expected value of series of uniformly converges random variables [duplicate]

Let $X_1,X_2,X_3,...$ a series of i.i.d. variables with $X_i \sim \mathcal{U}(0,1)$. Let $N=\inf\{n\mid \sum_{i=1}^{n}X_i\geq1\}$ Prove that $E(N)=e$. I don't really have a clue how to even start ...
2
votes
1answer
37 views

Why is this distribution Poissonian?

Do this experiment. Draw 10000 random number in $[0,1]$ according to the uniform distribution. Order them in the increasing order. The difference between two neighbouring numbers follows a Poisson ...
1
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0answers
30 views

If $\{X_n\}$ is a Cauchy seq of r.vs a.s., then why does it converge to some r.v?

I was suddenly suspicious of what title says. My logics are follows. $\{X_n\}$ is a Cauchy seq of r.vs $P$-a.s. $\Leftrightarrow$ $P(\{X_n\}$is a Cauchy $)=1$ $\Leftrightarrow$ $\exists ...
2
votes
1answer
68 views

Proving it's a martingale and more conditions.

Let $(X_{n})_{n>0}$ be a sequence of random variables in $[0, 1]$ and assuming that ($X_{0}=a) \epsilon [0, 1]$ then: $Pr\left(X_{n+1}=\frac{X_{n}}{2}|\mathcal{F}_{n}\right)=1-X_{n}$ and ...
0
votes
1answer
32 views

Find the distribution function F(y) [closed]

Can someone show me how to do this problem? Don't know how to format my work here.
0
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0answers
39 views

How to show $ P\big(\big|\frac{X}{n}-p\big|>a\big)\le\frac{\sqrt{p(1-p)}}{a^2n}min\big\{\sqrt{p(1-p)},a\sqrt{n}\big\}$

Let $X$ be binomial, $B(p,n)$ with $p>0$ fixed, and $a>0$. Show that, $\displaystyle ...
2
votes
2answers
103 views

Monte Carlo Importance Sampling

I am reading the book on Monte Carlo by Sobol (A Primer for the Monte Carlo Method). In the section on Importance Sampling, he writes: $I = \int_a^b g(x) \: dx$ "to compute this integral, we could ...
-3
votes
1answer
58 views

Find the expectation $E[X]$ [closed]

Let $X$ be a random variable which is uniformly chosen from the set of positive odd numbers less then 100. Find the expectation $E[X]$?
1
vote
1answer
25 views

A continuous random variable map by continuous function will become continuous?

Let $X$ be a continuous random variable and let $g$ be a non-constant real-valued continuous function. Prove or disprove that $g(X)$ is a continuous random variable. Note : Here it is assumed that ...
1
vote
3answers
70 views

Expected value of the function of a random variable

I am studying Probability and Monte Carlo methods, and it feels that the more I study the less I truly understand the theory. I guess I just confuse myself now. So the expected value of a random ...
1
vote
1answer
30 views

Showing independence of rectangular events…

Suppose I have a sequence of independent random variables $\{X_n, n \in \mathbb N\}$. How do I show formally that $P((X_1,...,X_n)\in A, (X_{n+1},...)\in B) = P((X_1,...,X_n)\in A)P((X_{n+1},...)\in ...
2
votes
1answer
33 views

Convergence almost surely and B-C lemmas

Showing the expectation is straightforward. I am not sure how to use the Borel-Cantelli lemmas to show the almost surely part.
2
votes
2answers
53 views

Computing cov of 2 binomial random variables

we drop a normal cube 20 times. X - is the number of even values Y - is the number of times the cube landed on 3. As much as I understand: $$X\sim B(20, \frac{1}{2}) \\Y \sim B(20, \frac{1}{6} )$$ ...
1
vote
0answers
19 views

Reference for higher moments of nonnegative random variable as integrals of the CDF

I know how to prove $E(X^n) = \int_0^\infty \! (1 -F_X(u^{1/n})) \, \mathrm{d} u$, for a positive and continuous random variable $X$ with CDF $F_X(x)$---note that for $n=1$ it is the standard $E(X) = ...
1
vote
1answer
35 views

Does every random variable(continous) has a probability density function?

what is the criterion for a random variable(continous) for existence of probability density function for it? Could you provide some cases of random variable(continous) where pdf ceases to exist.
0
votes
1answer
25 views

calculation of variance from cdf (no mathematical expression available)

Is it possible to calculate the variance of a continous random variable from the Cummulative distributive function plot ? We dont have the mathematical expression for cdf, all we have is just a plot ...
1
vote
3answers
39 views

Relationship between Binomial and Bernoulli?

How should I understand the difference or relationship between Binomial and Bernoulli distribution?
0
votes
0answers
46 views

What is the probability density function of the cosine of a gaussian random variable?

I want to find the probability density function of $Y=\cos(X)$, where $X\sim N(\mu, \sigma^2)$. The answer is known when $X$ is uniformly distributed $U(-\pi, \pi)$ and it is an arcsin pdf, given by, ...
2
votes
2answers
50 views

Independence of random sum variables

Let $(T_i)_{i \in \mathbb{N}}$ be a family of i.i.d. random variables where every $T_i \sim\mathrm{Exp}(\lambda)$. Now let $$Y :=\sum\limits_{j=1}^N T_j$$ such that for all $1 \leq j \leq N-1$ we have ...
2
votes
1answer
31 views

Question about $L^1$ convergence for random variables

For a random variable $X \colon \Omega \to \mathbb{R}$ and a sequence of random variables $X_n$ with $$ \lim_{n \to \infty} \mathbb{E} [|X_n -X|] = 0,$$ I have found that $$ \lim_{n\to \infty} ...
1
vote
0answers
26 views

Random sampling and i.i.d.

Can you help me to clarify the following concepts by stating whether what I have written below is right or wrong? -random sampling: units are drawn from the population with a known probability of ...
0
votes
1answer
42 views

subscript notation in conditional probability

$X$ and $Y$ are two discrete random variables with joint p.m.f $p_{XY}$ such that $p_{XY}(x_i,y_j) = P(X=x_i, Y=y_i)$. I came across a notation that refers to $p_{X}(x|y)$. How do I express it in the ...
0
votes
1answer
10 views

cumulative distribution function calculation 3

given FX(X) = x^2, compute P(1/4 < X < 1/2). sorry, I am new to here so don't really know how to type them more mathematically.