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

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0
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2answers
398 views

Poisson arrivals during an exponentially distributed interval

This is a marked homework question, so please try not to write complete solutions here: The number of customers that arrive at a service station during a time t is a Poisson random variable with ...
1
vote
1answer
15 views

Probability: Expectation: indicator RV, what is 1-((N-1)/N))^n?

Say there are N coupon types, you collect n coupons, and what's the expected number of types of coupons? My question is specifically about $1-(\frac{N-1}{N})^n$, the probability of getting a coupon ...
0
votes
1answer
26 views

Probability that there is sub-sequence of exact length

Can you help me to solve the following: Find probability that in sequence of N random uniformly distributed numbers there is increasing sub-sequence of exact length L.
4
votes
1answer
35 views

Almost surely vs expectation

Let $X_1, X_2, X_3 \dots$ be a sequence of random variables. In the limit as $i \rightarrow \infty$ we have $$ X_i \rightarrow 0 \text{ almost surely} $$ Does it follow that In the limit ...
1
vote
3answers
6k views

How to calculate the median of a continuous random variable

$X$ is a continuous random variable with probability density function $f(x)= \dfrac{2x}{15}$ where $1≤x≤4$. What is the median of X?
0
votes
1answer
21 views

The smallest filtration for which a sequence of random variables is adapted

Let $X_1, ..., X_n$ be a sequence of random variables. Show that $\hspace{60pt}$ $\mathcal{F}_n$ = $\sigma(X_1, ..., X_n)$ is the smallest filtration such that the sequence $X_1, ..., X_n$ is ...
3
votes
3answers
259 views

Random Variables that aren't measurable

I've been reading through a math. stats. book, and I'm a little confused with the concept of measurable random variables. The book states: Let $(E, \mathcal{E})$ and $(F,\mathcal{F})$ be two ...
0
votes
2answers
9 views

Showing that $n1_{ \lbrace U<1/n \rbrace}$ converges to $0$ almost surely

Let $U \sim \text{Uniform}[0, 1]$ and $X_n = n1_{\lbrace U< 1/n \rbrace}$. I want to show that $X_n$ converges to $0$ almost surely. My attempt: I use Fatou's Lemma with the reasoning that if I ...
2
votes
1answer
28 views

how to prove independence in this case

The question is : $X_1,...,X_n$ are i.i.d.$Uniform(0,\theta)$. Let $X_{(n)}$ denote the maximum of these $n$ random variables. Prove that $\frac{X_1}{X_{(n)}}$ and $X_{(n)}$ are independent. What I ...
0
votes
0answers
19 views

How to Justify the exclusion of some samples?

I am calculating the asymptotic cumulative distribution of $M_n = \max(X_1,X_2,\dots,X_N)$. My problem is $X_1,X_2,\dots X_p$ and $X_k,X_{k+1},\dots,X_N$ have non identical CDF for $p<<k$ and ...
3
votes
2answers
67 views

If $X_i$ are iid $U(0,1)$ random variables, $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$

I want to show $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$ as $n \to \infty$, where $X_i$ is an i.i.d sequence of $[0,1]$-uniformly distributed random ...
0
votes
1answer
16 views

Random Variable probability summation tweaking

I can't seem to figure out what they do to get to the bottom
0
votes
2answers
38 views

PDF of Gamma R.V. [on hold]

I know that $X \sim \exp(λ)$, $Z\sim \exp(λ)$ and $Y\sim \exp(λ)$ for $λ>0$. I also know that all three: $X, Y$ and $Z$ are independent. How do I find a pdf for $X+Z+Y$?
1
vote
1answer
64 views

Approximate normal distribution

Let $ X \sim N (0, 1)$. For $x$ large enough, the tail of the distribution of $X$ may be approximated as $$P(X > x) \sim e^{-x^2/2}/(x\sqrt{2\pi})$$ Consider a sequence of independent r.v. all ...
1
vote
1answer
12 views

Positive component of a submartingale is a submartingale

I am trying to prove the Doob's Upcrossing Lemma and the first step requires to prove that: If $X$ is a submartingale, then $(X-a)_+$ is a submartingale. I found it intuitive but i failed to prove. ...
0
votes
1answer
308 views

Continuous random variables word problem

I'm having difficulty with a word problem on continuous random variables. Would you be able to help? Here it is: If you're n minutes early for an appointment, then you are charged n times 3 dollars, ...
0
votes
1answer
12 views

Can I sum variances to a compound variance?

Say I have three locations A,B,C and I have a person going from A to B and measure the time it takes. Same for B to C. Let the variance of the time it takes for the path AB be a and for BC b. Is it ...
2
votes
3answers
43 views

Calculating the mean and variance of a distribution

Suppose $$P(x) = \frac{1}{\sqrt{2\pi\cdot 36}}e^{-\frac{1}{2}\cdot (\frac{x-2}{6})^2}$$ What is the mean of $X$? What is the standard deviation of $X$? Suppose $X$ has mean $4$ and variance $4$. ...
2
votes
2answers
41 views

Limit law of real-valued independent random variables

Let $X_n$ and $Y_n$ be real-valued independent r.vs, each of whose limit law is $X$ and Y, resp. i.e $X_n \overset{d}{\to} X$ and $Y_n \overset{d}{\to} Y$ for some r.vs $X$ and $Y$. Then, are $X$ ...
0
votes
1answer
13 views

Independent random variables convergence almost surely

I'm preparing for a final exam and I'm working through a practice exam. I'm somewhat stuck on a problem. The random variables $\{X_i\}$ are all independent and all satisfy $E[X_i^4]\leq 1.0$, but ...
0
votes
1answer
25 views

Approximate as Independent Identically distributed

If $N$ random variables are identically distributed but weakly correlated, in what condition we can approximate them as independent identically distributed (iid) ? I saw an old paper where based on ...
-2
votes
1answer
40 views

$Var(X)< \infty \implies E|X| < \infty$?

$\textrm{Var}(X)<\infty$ implies $E|X|<\infty$ ? And what about the converse?
0
votes
0answers
14 views

How to change 1D Metropolis into 2D?

I've written a MATLAB function to generate random numbers from a given univariate distribution using the Metorpolis algorithm. Here it is: ...
3
votes
1answer
37 views

A property about tail equivalent random variables

Let $(X_n)$ and $(Y_n)$ be tail equivalent random variables i.e. $\sum_{i=1}^{\infty}\mathbb P(X_i\neq Y_i)<\infty$ Show that $\sum_{n=1}^{\infty}X_n$ and $\sum_{n=1}^{\infty}Y_n$ converge or ...
2
votes
2answers
44 views

$X:\Omega \to \mathbb{N}$ is random variable, How to prove that $E[x]=\sum_{i} \Pr(X\ge i)$?

I'm stuck with this proof: $X:\Omega \to \mathbb{N}$ is random variable, prove that $\mathbb{E}[X]=\sum_{i=1,2,3...} \mathbb{P}(X\ge i)$? How I'm proving it? I'm starting with the definition: ...
0
votes
2answers
832 views

What is the expected number of times a 6 appears when a fair die is rolled 10 times?

Ok, so I think I have a working solution to this problem. Heres how I would solve it: so you look at a 6 appearing as a success and everything else as a failure. So from here you can you use the ...
1
vote
2answers
37 views

Sum over product of two binomial distributions

The problem is that of a two-stage "binomial experiment", where first a number $k$ out of $n$ is drawn (each element with probability $p_1$) and later a number $m$ out of those $k$ is drawn (each ...
2
votes
1answer
44 views

Difference between two independent binomial random variables with equal success probability

Let $X$ ~ $Bin(n,p)$ and $Y$ ~ $Bin(m,p)$ be two independent random variables. Find the distribution of $Z=X-Y$. see also Difference of two binomial random variables I figured this out: $$ ...
1
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1answer
49 views

We roll a standard fair die over and over. What is the expected number of rolls until the first pair of consecutive sixes appears.

During class we split this task into smaller pieces: Let $X$ = r.v. denoting the result of the first roll Let $Y_1$ = r.v. denoting the result of the first roll Let $Y_2$ = r.v. denoting the result ...
-1
votes
2answers
37 views

The product of a normal and Bernoulli variables, independent from each other

Let $X\sim N(0,1)$ and let $Z$ be a random variable independent of $X$ such that: \begin{equation} \Pr(Z=z) = \begin{cases} \frac{1}{2} & \mbox{if $z = -1$ or $z=1$}, \\ \\ 0 & ...
0
votes
2answers
74 views

How to apply Chebyshev's inequality? [closed]

When a fair coin is tossed 16 times, the random number of tails, $X$, has mean $\mathbb{E}[X]=8$ and variance $\operatorname{Var}(X)=4$. Using Chebyshev's Inequality, determine a lower bound for ...
0
votes
1answer
31 views

Binomial distribution, when variable isn't x

I've been using the formula $$p(x,N)=\frac{N!}{(\frac{N+x}{2})!(\frac{N-x}{2})!} p^{1/2(N+x)} q^{1/2(N-x)}$$ to determine the probability for a dog who walks in a straight line and can either move ...
1
vote
1answer
369 views

Given x is an exponential random variable, find median & probability

For the median, I believe that I should integrate the function, ∫x0λe−λtdt=1−e−λx Then I need 1−e−λm=.5 for m, which is equivalent to e−λm=.5. m=ln(2)/λ =>m=ln(2)/.2
1
vote
1answer
15 views

Suppose a random variable X has mean 0 and moment generating function as follows, find values of a and b

$M_x(t)=a(1+e^{-2t}+e^{-t} +e^t+be^{2t}), -\infty<t<\infty$ Do I take the first derivative of this function? How do I solve for two variables given only one equation? And as a followup ...
1
vote
1answer
57 views

Z-transform of white noise

Let's say we have a discrete-time white noise with variance $\sigma ^2$ that is we have a zero-mean Gaussian random variable with variance $\sigma ^2$. What is the $Z$ transform of this noise? Can we ...
1
vote
2answers
23 views

Coin Tossing Conditional Probability

On a practice test with no available solutions I was asked the following two-part question: 1) If a coin is tossed until three consecutive heads are shown, what is the probability that one tail is ...
3
votes
3answers
48 views

Independence of two normally distributed random variables

Let $X \sim \mathcal{N}(0, 1)$ and $Y$ be a random variable independent of $X$ such that \begin{align*} P(Y=y) = \begin{cases} \frac{1}{2} & y = -1\\ \frac{1}{2} & y = 1\\ 0 & ...
1
vote
1answer
23 views

A question about Poisson process such that…

I got the following problem: Suppose that instances of some event occur in accordance with a Poisson process having a rate of 24 instances an hour Suppose we take a time-interval of length 1 hour ...
1
vote
1answer
51 views

Convegence in Probability but not almost surely

Let $ {a_n} $ be a sequence of numbers in $ (0,1) $ such that $ a_n\to0 $, but $\sum_{i=0}^\infty a_n=\infty$. Suppose $X_1, X_2, \ldots$ be independent random variables with ...
2
votes
1answer
35 views

A question about different pairs that are formed from a set of 16 different poeple such that…

I got the following problem: Given a set of 16 different people. We partition the people into pairs of two. Each pair needs to accomplish a task. And the probability that a pair accomplishes ...
0
votes
1answer
271 views

Continuous random variable with density function

$X$ is a continuous random variable with p.d.f. $f(x) = c\space e^{-x} ; x > 1$. Find $P(X < 3|X > 2)$
1
vote
1answer
52 views

Poisson distribution and idd random variables - Proof of an equality

I want to solve the following task The first part was very easy for me. But I dont know how to solve the second one. I guess I even understood it completely. I am thankful for any kind of help!
0
votes
2answers
32 views

Product of a Continuous and Discrete Distribution

Let $X \sim \mathcal{N}(0, 1)$ and $Y$ be a random variable independent of $X$ such that \begin{align*} P(Y=y) = \begin{cases} \frac{1}{2} & y = -1\\ \frac{1}{2} & y = 1\\ 0 & ...
0
votes
1answer
48 views

Mixed Distributions - Expectation and Variance

A bike has probability of breaking down $p$, on any given day. The repair cost of the bike, whenever it breaks down, is distributed as a Gamma random variable with shape $\alpha$ and rate ...
0
votes
0answers
20 views

Geometric Distributions of Random Variables

Suppose that $X_1, X_2, ...$ are random variables that are independent and geometric, although perhaps with different parameters. Find as many of the following as is feasible: $X_1+X_2, X_1+X_2+X_3$, ...
0
votes
0answers
17 views

Product of two random variables - Resulting Distribution and Correlation?

Let $X \sim \mathcal{N}(0,1)$ and let $Z$ be a random variable independent of $X$ such that \begin{align*} P(Z=z) = \begin{cases}\frac{1}{2} & z=-1\\ \frac{1}{2} & z = 1\\ 0 & ...
2
votes
4answers
27 views

sum and difference between two independent Poisson random variables [closed]

Let $X$ and $Y$ be independent Poisson variables with respective parameters $a$ and $b$. What is the distribution function of $X+Y$? the conditional distribution of $X$, given $X+Y=n$? same, ...
0
votes
1answer
15 views

Find distribution function $F_Y(y)$ of random variable $Y$ [closed]

Tomorrow an midterm exam so I really need your help Let X be a random variable uniformly distributed in $[-1, 4]$. Say $Y = |X|$. Calculate the distribution function $F_y(y)$ of the random ...
1
vote
1answer
26 views

Expectation values of powers of a Poisson random variable

The expectation values for integer powers of a Poisson random variable $X\sim Poiss(\lambda)$ are well known. I'm interested in the expectation value of $X^\alpha$ for arbitrary rational $\alpha$. ...
0
votes
0answers
17 views

Find Cumulative and the probability density function of Y

Usually I would integrate the function $y=x^2$ from 2 to 1 and to find the probability density function but I need to show it in terms of t. How do I do this? Also is the cumulative distribution = ...