0
votes
2answers
24 views

Probability Density Function with continuous random variables

Let $X$ have density $$ f_X(x) = \begin{cases} \sqrt{3(x+2)}/6 & -2 \leq x \leq 1 \\ 0 & \text{otherwise}. \end{cases} $$ Find the probability that $X$ is positive. Would this just ...
0
votes
1answer
21 views

calculating variance of a random variable

Suppose you have a playlist consisting of four songs that you play in a smart shuffle mode. In this mode, after the current song is played, the next song is chosen randomly from the other three ...
1
vote
1answer
32 views

Calculating bounds with multiple random variables.

I have this problem: Suppose there are 4 students (who we'll refer to as A, B, C, and D) in a class and each student is equally likely to have been born in any of the twelve months of the year. For ...
0
votes
1answer
24 views

calculating X, Y, Z random variables

Suppose X, Y, and Z are random variables that each take the value 0 or 1. If P(X=0,Y=1,Z=0)=1/3 and P(X=0,Y=1,Z=1)=1/4, what is the value of P(X=0,Y=1)? I am trying to calculate this but I am really ...
0
votes
2answers
23 views

Covariance of dependent random variables from a Poisson process

Question: Given a Poisson process $N(t),t≥0$ with rate $λ$, calculate the covariance of $N(2)$ and $N(3)$. Attempt: So clearly $N(2) \sim Po(2\lambda)$ and $N(3) \sim Po(3\lambda)$. So, ...
0
votes
1answer
35 views

probability of playing music player on shuffle and listening to every song.

I have a few problems I am trying to work out but I am not totally confident in my answers: The problem is such: Suppose you have a playlist consisting of four songs. You play your playlist in ...
1
vote
1answer
26 views

Finding density function of random variable, which is division of two other random variables.

I have following 2-dimensional random variable $(x,y)$: $$ f(x,y) = 1, \quad 0 \leq x \leq 1, \quad 0 < y \leq 1 $$ I have to find density function of random variable $Z = \frac{X}{Y}$. I am ...
1
vote
2answers
71 views

Random Variable Probabilities

Suppose you have a playlist consisting of four songs that you play in a smart shuffle mode. In this mode, after the current song is played, the next song is chosen randomly from the other four ...
2
votes
1answer
33 views

Probability exercise Bernoulli. [closed]

Probability random signals. Im late I have no idea to start and this is for tomorrow. I was on training and have no break to do this work. I do this.You are an Internet savvy and enjoy watching video ...
1
vote
3answers
30 views

Probability excersice

If $Z$ is a Gaussian random variable with mean $\mu_Z = 0$ and variance $\sigma^2_Z = 1$, and $Y$ is defined as: $$Y=a + bZ +cZ^2$$ for some constants $a, b, c$ show that the correlation ...
1
vote
0answers
24 views

Extension to Classical Coupon Collectors Problem

If there's n different coupons. Instead of ordering coupons one-by-one until you collect all n coupons as in the traditional 'Coupon Collector Problem', what if the coupons came in packs of m coupons. ...
0
votes
0answers
25 views

Extension to the Coupon Collector Problem

If there's n different coupons. Instead of ordering coupons one-by-one until you collect all n coupons as in the traditional 'Coupon Collector Problem', what if the coupons came in packs of m coupons. ...
0
votes
1answer
26 views

Calculate $E[XY]$ of dependent variables

I'm having a little trouble whit a probabilistic exercise. The problem says this: There's a vase whit 10 marbles, 4 black and 6 white. Now I extract 2 of them without reposition. Being $X,Y$ random ...
5
votes
0answers
56 views

Covergenge of the sum of reciprocal random variable.

If $(X_n)_{n\in\mathbb{N}}$ are independent identically distributed random variables with density $f$ even, continuous in $0$ and such that $f(0)>0$, then $$\frac{1}{n}\left(\frac{1}{X_1}+\dots + ...
1
vote
0answers
26 views

Conditional probability of function of two RVs

I have two random variables, $X, Y$ and their joint pdf, $f_{XY}(x,y )$. I am able to find the marginal PDFs, $f_X(x)$ and $f_Y(y)$ using $f_X(x) = \int_{-\infty}^{\infty}f_{XY}(x,y)dy$ and similar ...
0
votes
1answer
66 views

Roll a 6-sided fair die until a 6 appears. Let X = the number of 1's that are rolled. Find Var(X).

Let X = the number of 1's that are rolled. Find E[X] and Var(X). I can't seem to calculate Var(X). I've calculated E[X] = 1. I let R = the number of non-6 rolls, and I let Y = the number of rolls ...
3
votes
1answer
39 views

probability of the sum of i.i.d. RV with uniform distribution being $>x$

I am solving a question for applied stochastic processes homework and I am stuck on this part: Let $X_1,X_2,\cdots, X_n$ be independent identically distributed random variables with uniform ...
1
vote
2answers
37 views

Function of random variable

I have this question: Suppose P(X=0)=1/2 and P(X=8)=1/2. What's the value of E[Y] if Y=(X^2)? So I am having trouble understanding how to go about doing this ...
0
votes
4answers
72 views

Expected value of rolling dice until getting a $3$

I am having trouble with this question with regards to random variables and calculating expected values: Suppose I keep tossing a fair six-sided dice until I roll a $3$. Let $X$ be the number of ...
0
votes
2answers
38 views

Expected value of random variable

I have this question: What's the expected value of a random variable $X$ if $P(X=1)=1/3$, $P(X=2)=1/3$, and $P(X=6)=1/3$? I am very confused as to how I can work this problem out. I was thinking ...
0
votes
2answers
22 views

existence of a RV with distribution given by a linear combination of other distributions

Question: Let $X$ and $Y$ be random variables defined on a $(\Omega,\mathfrak{F},\mathbb{P})$ probability space with distribution functions $F_X(t)$ and $F_Y(t)$, respectively. (a) Show that for any ...
1
vote
2answers
19 views

Show that $Cov(X,Y) \geq -23$

if $X,Y$ are two random variables and: $Var(X) = Var( Y) = 23$ how can i show that $Cov(X,Y)\geq -23$ can someone give me some hints on how to show it?(not an answer) i know that $Cov(X,Y) = E(XY) ...
0
votes
1answer
30 views

Probability of sum of two continuous is greater than 1

I am given a two-dimensional absolute continuous random variable, whose density function is defined as followed: $f_X,_Y(x,y)=1/2 $ if $0<x<1$ and $0<y<4x$. I have found the marginal ...
0
votes
2answers
18 views

Finding Y's marginal distribution where joint distribution of $f_{X,Y}(x,y) = 1/2$ in $0 < x < 1$ and $0 < y < 4x$

I am given a two-dimensional vector (X,Y) whose joint density function is as follows: $f_{X,Y}(x,y)=1/2$ if $0<x<$ 1 and $0<y<4x$. I am now to find the marginal densities of X and Y. I ...
0
votes
0answers
18 views

Find the smallest n satisfying sum of variance is < 0.01

I'm stumped on how to do this exercise. There are $X_1,...,X_n$ different random variables who all have the same distribution and are independent. The variance of any given $X_i$ is known to be ...
1
vote
0answers
34 views

I want to show $\phi_{X}(a_{1},a_{2},\cdots,a_{n})=\prod_{i=1}^{n}\phi_{X_{i}}(a_{i})$

Let $n \in \mathbb N$ and $X$ be an $\mathbb R^n$ valued random variable on $(\Omega ,\mathcal F,P)$ Define its characteristic function to be $$\phi_{X}(a)=E(e^{i\langle X,a\rangle})$$ where $a \in ...
1
vote
1answer
35 views

Application of the Dominated Convergence Theorem (probabilistic version).

I am currently working on the following problem and I think I've got the solution more or less, but there is a minor question about the usage of the Dominated Convergence Theorem. Let $f: [0,1] ...
1
vote
1answer
38 views

Find the random variable, value function, and value you would pay to break even…

In a game you receive three cards, $\omega$ , from a well-shuffled deck. You then receive $10 if the hand contains at least two face cards. In order to determine how much you would be willing to pay, ...
1
vote
1answer
87 views

Convergence in probability of product and division of two random variables

How can I prove the following: Let $X_i$ and $Y_i$, $i = 1, \ldots, n$, $X$ and $Y$ be random variables defined on the probability space $(\Omega, \mathcal F, \mathbb P)$ and assume that $X_n$ ...
1
vote
1answer
29 views

Confidence interval and normal distribution

For question (a), is the answer 0.7143? For question (b), is the answer 10.85 and 11.95 ?
1
vote
1answer
259 views

Probability: Normal Distribution

Each item produced by a certain manufacturer is, independently, of acceptable quality with probability $0.95$. Approximate the probability (by a normal distribution) that at most $10$ of the ...
0
votes
1answer
72 views

Show that $\lim\limits_{n\rightarrow\infty} e^{-n}\sum\limits_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$

Show that $\displaystyle\lim_{n\rightarrow\infty} e^{-n}\sum_{k=0}^n \frac{n^k}{k!}=\frac{1}{2}$ using the fact that if $X_j$ are independent and identically distributed as Poisson(1), and ...
1
vote
1answer
161 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
3
votes
0answers
117 views

Show that $(X_{n},Y) \to^{\mathcal{D}} (X,Y)$ AND if $X=h(Y)$ where $h$ is a Borel function that $X_{n}\to^{P} X$

Let $X_{n}$, $X$, and $Y$ be real-valued r.v.'s all defined on the same space $(\Omega, \mathcal{A},\mathbb P)$. Assume that $\lim_{n \to \infty}\mathbb E\{f(X_{n})g(Y)\}=\mathbb E\{f(X)g(Y)\}$ ...
3
votes
2answers
69 views

$\operatorname{Bin}{(n,U)}$, where $U$ is uniform on $(0,1)$

A question in my probability class: Let $X$ have the binomial distribution $\operatorname{Bin}{(n,U)}$, where $U$ is uniform on $(0,1)$. Show that $X$ is uniformly distributed on $\{0,1,\dotsc, n\}$. ...
0
votes
1answer
94 views

Random Variable Problem w/ variance

Three zero mean, unit variance random variables X, Y, and Z are added to form a new random variable, W = X + Y + Z. Random variables X and Y are uncorrelated, X and Z have a correlation coefficient of ...
1
vote
1answer
42 views

What is the variance of this random variable?

A clerk drops $n$ matching pairs of letters and envelopes. He then places the letters into the envelopes in a random order. Let $X$ be the number of correctly matched pairs. Find the variance of $X$.
2
votes
1answer
37 views

$X$ is standard normal if $X=Y1_{\{|Y|\le a\}}-Y1_{\{|Y|>a\}}$ where $Y$ is standard normal

$X$ is standard normal if $X=Y1_{\{|Y|\le a\}}-Y1_{\{|Y|>a\}}$ where $Y$ is standard normal. $$F_X(x)=P(X\le x)=P(\{Y\le x\}\cap\{|Y|\le a\})+P(\{-Y\le x\}\cap\{|Y|> a\})$$ How can I simplify ...
0
votes
1answer
35 views

sum of independent random variables where $N$ is a random variable

I want to show $E[S_N]=E[N]E[X_j]$ where: $X_1,X_2,\ldots$ is a sequence of independent random variables, and $N$ is a random variable independent of the sequence. $S_n=\sum_{i=1}^n X_i$, ...
1
vote
1answer
50 views

Solution to Billinglsey (1995) problem 20.22

Let $Y_1\leq Y_2\leq ...$ be random variables s.t. $\mathrm{plim} Y_n = Y$. Show that $Y_n \to Y$ with probability 1. Some hints? My strategy would be to prove that $\sum P(\lvert Y_n -Y \rvert > ...
1
vote
2answers
34 views

Heteroskadasticity and Linear Probability

Question Suppose $(Y,X,U)$ be a random vector such that $$ Y = X'\beta + U. $$ Suppose $Y$ takes values in $\{0,1\}$ and that $E[Y\mid X] = X'\beta$. Is it reasonable to assume that $Var[U\mid X]$ ...
1
vote
1answer
88 views

Upper bound on sum of i.i.d. random variables

Here's a problem I've been struggling with: Let $X_1, X_2, X_3, \ldots$ be an i.i.d. sequence of random variables with finite moment generating function $M(t)$. Define the sum $S_n = X_1 + \ldots ...
1
vote
1answer
60 views

Maximising Entropy of Random Variable taking Positive Integer Values

A random variable $X$ takes positive integer values and $E[X]=6$. What distribution of the random variable $X$ maximises the entropy $H(X)$? What if $X$ can only take a finite number of values? ...
2
votes
1answer
100 views

Exponential distribution: Finding the parameter

Please help me solve the following problem Time of production of one electronic component is given with exponential distribution with parameter λ. If the process lasts less than 3 hours, the ...
1
vote
0answers
57 views

Copulas/Probability Theory

So I have a basic understanding of copulas but wanted to verify I'm applying things correctly to reach the correct outcomes.. Show that as $\theta\to\infty, C^{Fr}(u_1,u_2)\to\min(u_1,u_2)$, the ...
0
votes
1answer
25 views

Please help with this probabilities

The daily production of a factory is 20 articles, of which two are always defect. A sample of four is taken. Let X be the random variable that assigns the number of defect articles in the sample. ...
2
votes
1answer
132 views

Showing certain functions are random variables

Assume $\{X_k\}_{k \in \mathbb N}$ are random variables on a probability space. Define induced random walk by $S_0 = 0$ and $S_k = \sum_{i=1}^{k}X_i$. Now let $n = \inf\{p > 0: S_p > 0\}$ be ...
1
vote
1answer
102 views

show it is a random variable

Let X and Y be random variables and let A be an event. Show that the function $$Z(\omega)=\begin{cases}X(\omega) \quad \text{if} \; \omega \in A\\ Y(\omega) \quad \text{if} \; \omega \in A^c ...
1
vote
2answers
466 views

Solving for the covariance of a joint pdf

Let X and Y have a joint pdf given by $f_{x,y}(x,y) = \begin{cases} 1 & \text{if } 0<y<1,\text{ } y-1<x<1-y \\ 0 & \text{otherwise} \end{cases}$. (a) Find Cov(X,Y) and ...
2
votes
0answers
56 views

Expectation of random variables

a) Show that $E\{X-E(X)\} = 0$ for any random variable $X$. b) Use the result in part (a) and the following equation to show that if two random variables are independent then they are uncorrelated, If ...