0
votes
2answers
39 views

Probabilities and random variable problem

Suppose we have a random variable X, and we are given the numerical values of its expectation as well as its s.d. (standard deviation). How can I go about finding the maximum value the probability of ...
0
votes
3answers
30 views

Generate random variable from series of its expected values E[X], E[X^2], E[X^3], …?

Given a series of all the expected values of a random variable, can we find the random variable itself ?
1
vote
1answer
19 views

Find the cdf associated with each pdf (NOT transformation)

Find the cdc associated with each pdf: a) f(x) = 3(1-x)^2 , 0 < x < 1 , zero elsewhere b) f(x) = 1/x^2 , -infinity < x < infinity The answers are a) 1-(1-x)^3 , 0 <= x < 1 b) 1 ...
0
votes
1answer
36 views

Variational problem concerning variances

Let $\phi$ be the family consisting of all random variables $X$ such that $P(X\in [0,1])=1$, $EX=\frac{1}{3}$, $P(X<\frac{1}{4})<\frac{1}{2}$, $P(X>\frac{1}{4})\geq\frac{1}{2}$. Calculate ...
1
vote
1answer
21 views

Finding the CDF of $g(X)$ where $X$ is a continuous random variable

I imagine this is a rather simple problem, but I'm having a bit of a hard time actually finding the answer. $X \sim \mathrm{Exp}(0.2)$ and $W=g(X)$ given by $g(X) = \begin{cases} X^{\frac{1}{3}} ...
1
vote
1answer
41 views

A fair die is tossed until the sequence “44” is seen. Let N be the number of tosses this requires. Find E $N$

A fair die is tossed until the sequence “44” is seen. Let $N$ be the number of tosses this requires. Find $E[N]$. I have my own solution which I need someone to verify. And this problem has to be ...
1
vote
2answers
35 views

Continuous and Discrete random variable distribution function

I have a very basic question in probability. It pertains to the difference between a continuous random variable distribution function and a discrete one. This question has confused me many times. ...
0
votes
0answers
6 views

Methods for Uncorrelating data - Comparison

I see that both PCA and Cholesky Decomposition could be used for uncorrelating correlated data. When should one be used? What are the assumptions made by each model. When do the methods fail? Are ...
2
votes
1answer
25 views

Joint density function with absolute value

Let X and Y have joint density fXY (x, y) = kxy^2 where j0 ≤ x, y ≤ 1, 0 otherwise. Compute Pr(|X − Y | < 0.5). So I found that k=6, but can't figure out the probability part after working on ...
1
vote
2answers
14 views

PMF of X: Number of trials to draw a chip

Let a bowl contain 10 chips of the same size and shape. One and only one of these chips is red. Continue to draw chips from the bowl, one at a time and at random and without replacement, until the red ...
0
votes
2answers
43 views

PMF of number of heads of 4 coin tosses

Let X equal the number of heads in four independent flips of a coin. Using certain assumptions, determine the pmf of X and compute the probability that X is equal to an odd number. I initially ...
0
votes
2answers
26 views

If $P(X \geq k) = p^k$, for $k=0, 1, 2,…$ then $P(X=k)=p^k(1-p)$

If $P(X \geq k) = p^k$, for $k=0, 1, 2,...$ then $P(X=k)=p^k(1-p)$ The converse is immediate but I don't know how to approach the direct implication.
0
votes
1answer
18 views

Transformation of Random Variable results in strange CDF

I'm trying to transform a RV according to $Y=X^{-a}$ with $a>0$ and X being uniformly distributed in $[0,A]$: $ F_X(x) = \begin{cases}0 & x<0 \\ x/A & 0\leq x \leq A \\ 1 ...
0
votes
1answer
44 views

Random variable modeling arthroscopic meniscal repair

The below problem is from my introductory stats textbook, the chapter on random variables and probability distributions. I don't even know what's being asked, much less how to answer it. Any clues? ...
-1
votes
1answer
32 views

finding out the probability density of a random process

I have to find out the probability density function of a random process with the following specifications:z(t)= xcos(wt)-ysin(wt) where x and y are two independent gaussian random variables. Now what ...
3
votes
2answers
55 views

$X,Y$ independent then $X+Y$, $X-Y$ independent as well?

My question is simple: If $X$, $Y$ are independent random variables then $X+Y$, $X-Y$ independent as well?
-1
votes
3answers
37 views

Mean and Variance of Y using Expectation Operator

Let $$ Y =\sum_{k=1}^N a_kX_k $$ be the weighted sum of N independent random variables, $ X_k, k = 1, ... , N $ , each having mean $ \mu _{X_i} $ and variance $ \sigma ^2_{X_i} $. The weights $a_k$ ...
1
vote
2answers
19 views

Expectation of the min of three uniform i.i.d variables [closed]

Let it be T=min(X,Y,Z) where X,Y,Z ~ U(a,b). What is E(T)=?
5
votes
0answers
62 views

Probability that a five is seen before any of the even numbers are seen

A fair die is repeatedly tossed. What is the probability that a five is seen before any of the even numbers are seen? I have my own solution below and just want someone to verify it. According ...
-1
votes
2answers
41 views

How to work with the mode of a probability mass function

How do you work with a probability mass function in determining stuff related to the mode. Here's the question I have $P(X=x) = {\theta^n}{{n}\choose{x}}({\frac{1-\theta}{\theta}})^x, x = ...
1
vote
3answers
35 views

$X$ and $Y$ are independent and follow $U(0,1)$. Show $P(f(X) > Y) = \int_0^1 f(x) dx$

Let $X$ and $Y$ be two independent uniformly distributed r.v. on $[0,1]$, and $f$ is a continuous function from $[0,1]$ to $[0,1]$. Show that $P(f(X) > Y) = \int_0^1 f(x) dx$. I tried to prove ...
2
votes
0answers
45 views

Probability problem related to discrete random variables, binomial distribution.

I've just solved an exercise related to discrete random variables and maybe to the binomial distribution as well. I would like to know if my solution is correct, so here goes the problem statement ...
1
vote
1answer
31 views

Conditional expectation of symmetric Sigma algebra

Another exercise with conditional expectation that I have problems with. Let $\Omega=[-1,1]$, $\mathcal{F}=\mathcal{B}(\Omega)$, $\mathbb{P}=\frac{1}{2}\lambda$. Let X be a ...
2
votes
2answers
81 views

Expected distance between two vectors that belong to two different Gaussian distributions

Let $X$, $Y$ be two random variables that follow the Gaussian distributions with mean vectors $\mu_x$, $\mu_y$, and covariance matrices $\Sigma_x$, $\Sigma_y$, respectively. The probability density ...
1
vote
2answers
29 views

Prove Kolmogorov's SLLN by martingale.

Suppose $\xi_i$ are i.i.d. and $\mathbb E(|\xi_1|)\lt\infty$ Let $X_n=\sum_{i=1}^n\xi_i$ Then we have $\frac{X_n}{n}\to \mathbb E(|\xi_1|) $a.s. In the proof of this theorem: ...
1
vote
0answers
13 views

Simple random walk conditioning on non-return

Consider a simple symmetric random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$, with $S_0=0$. Let $P_{k,j}$ be the probability that the walker hits the point $k$ without returning to the origin in ...
0
votes
1answer
45 views

Expectation of sum of $n$ random variables.

Let $X_1, X_2, X_3 \ldots, X_n$ be n random variables which take values from $\{+1,-1\}$ uniformly. Let $S_n$ be defined as $$S_n =|X_1+ X_2+ X_3+ \dots +X_n| $$ Find the expectation $E(S_n)$ of the ...
0
votes
1answer
55 views

The probability of having $k$ successes before $r$ failures in a sequence of independent Bernoulli trials

Problem Find the probability of having $k$ successes before $r$ failures in a sequence of independent Bernoulli trials with $p$ being the probability of success. I thought of using the Binomial ...
1
vote
1answer
47 views

The probability of hitting a target at least twice, conditioned on hitting it at least once

I am working on some exercises related to random variables, since the subject is new to me, I feel a little insecure about my answers, I'll write the problem I've worked on to check if my solution is ...
0
votes
0answers
25 views

A R.V. is approximated as normal distribution by CLT. Can I use CLT again if I want to sum it with another R.V.?

My problem is, I have many independent normal random variables and since I need to have their squared distribution, I use Central Limit Theory to approximate the sum of its squared distribution as ...
1
vote
2answers
33 views

Deriving exponential distribution from geometric

Let $\lambda$ be the expected number of events in a unit time interval $[s,s+1]$ (events are independent of each other and of the time interval), and $T$ a continuous random variable that represents ...
1
vote
2answers
46 views

Showing that $\mathbb{E}|X\ln X| < \infty$ and $\mathbb{E}Z = \mathbb{E} X \ln X$ for given PDF of $Z$.

$X$ is real random variable such that $\mathbb{P}(X > 0) = 1$, $\mathbb{E}X^2 < \infty$, $\mathbb{E}X=1$. Let $Z$ be real random variable such that $\mathbb{P}(Z \in ...
0
votes
3answers
28 views

Variance of the sum of sample means

Let $X$ be a random variable with normal distribution with mean $ \theta$ and variance $ a>0$. Let $ Y $ be a random, variable with normal distribution with mean $\theta$ and variance $b>0$. ...
0
votes
1answer
33 views

Is the set $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ equal to $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$?

Difference between $\limsup\limits_n\{\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ and $\{\limsup\limits_n\frac{X_n}{\log(n)}>\frac{1}{\lambda}\}$ are the sets equal ? I think they would ...
0
votes
1answer
58 views

What is the mean and variance of $Y$, where $Y$ is sum of iid's

Here's my work for part a. I could use clarification on part b and d. Is part d the same as part a ($E[A_n] = E[Y]$) ? a) $$E[Y_n] = E[\frac{X_n}{2^n}]$$ ($X$'s are iid so...) $$= \frac{E[X]}{2^n} ...
2
votes
1answer
75 views

Mean of Piecewise function resting on IID random variables

Suppose IID random variables $X_t \sim X$ with support on $[0,1]$ and continuous CDF $F(\cdot)$. I wish to compute the expected value (mean) of the a piecewise function with form $$ \Phi (x,\mu) = ...
3
votes
3answers
51 views

Transformation(?) of Random Variables

There are two independent Gaussian R.Vs: $U:N(-1,1)$ and $V:N(1,1)$ How do I go about finding the PDF of the following transformations? X = U+V T = (U+2V, U-2V) W = U (with 50% chance), V (with ...
3
votes
2answers
81 views

How to give rigorous proofs of these two limit statements?

Let $X$ be a random variable with cumulative distribution function $F(x)$. Then how to rigorously prove the following two limit statements? $\lim_{x \to - \infty} F(x) = 0$. $\lim_{x \to + \infty} ...
-2
votes
1answer
35 views

Mean of max vs max of mean

If I have say an $n$ collection of 10 random variables $X_1, \ldots, X_{10}$ (so an $n \times 10$ matrix of values) from some underlying distribution whether Gaussian or uniform, and I calculate ...
0
votes
0answers
33 views

About independent random variables

Suppose that $\{X_n\}_{n\in\mathbb N}$ are identical distribued and independent random variables with values in $\mathbb Z$. I don't understand why ...
5
votes
0answers
52 views

Variational formulations in group theory?

I apologise if this is a naïve question. Are there any known / widely applicable / important variational formulations in (finite) group theory? That is, a relationship of the form $$\alpha(G) = ...
1
vote
2answers
33 views

Identifying the distribution which represents a negative binomial distribution as a compound poisson distribution

Suppose that the random variable $X$, which has a negative binomial distribution with probability $p$ and parameter $r$, can be represented as the summation of $N$ iid random variables $Y_1, Y_2, ...
1
vote
0answers
33 views

Probability distribution of k consecutive successes with n maximum trials

Let $X$ be a random variable that represents the number of trials of a given experiment. The outcome of a single trial is a Bernoulli random variable, with probability of success $p$, and trials are ...
0
votes
1answer
43 views

Expected value and variance of random process

Let $U,V$ be random variables with distributions $\mathcal{U}(-1,1)$ ,$\mathcal{E}(2)$ (uniform and exponential). If $U$ and $V$ are independent what is the variance and expectation of the random ...
1
vote
3answers
57 views

Are the absolute values of random variables iid if the random variables are iid?

If $X$ and $Y$ are independent and identically distributed (iid) random variables, does it imply that $|X|$ and $|Y|$ are iid? How would you go about proving this?
0
votes
1answer
18 views

What is the variance of multiple indicator random variables?!

Consider the following independent random variables $(V_1,V_2,V_3,\ldots,V_n)$ and a random variable $X$ as a function of these other random variables defined as follow on a set $A=(-\infty,x]$: $$ \ ...
0
votes
2answers
36 views

Marginal distributions of a random vector

I have the random vector $(X,Y)$ with density function $8x^{2}y$ for $0 < x < 1$, $0 < y < \sqrt{x}$ I am trying to find the marginal distributions of $X$ and $Y$. For $X$ this seems to be ...
0
votes
0answers
41 views

Sum of independent discrete random variable

Here is my attempt of deriving the sum of independent random variable in the discrete case : $\underline{\textbf{Sum of independent random variables}}$ Let $\mathcal{C_1}, \mathcal{C_2}$ be ...
1
vote
1answer
35 views

Bolzano–Weierstrass theorem for random variables?

I am wondering if there is something similar to the Bolzano–Weierstrass theorem for random sequences. Namely, let $\{x_n\}$ be a bounded random sequence. Is it true that, under some reasonable ...
0
votes
1answer
18 views

Show $P(X=n)=\left(\frac{1}{2}\right)^{n+1}$ for Poisson variable with exponentially distributed $\lambda$

I'm supposed to do the following, any help/pointer is appreciated: Suppose $X$ is Poisson distributed with mean $\lambda$. Suppose $\lambda$ is exponentially distributed with mean $1$. Show that ...