# Tagged Questions

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 ...
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 ...
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 ...
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 ...
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, ...
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 ...
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 ...
71 views

### Random Variable Probabilities

Suppose you have a playlist consisting of four songs that you play in a smart shuffle mode. In this mode, after the current song is played, the next song is chosen randomly from the other four ...
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 ...
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 ...
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. ...
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. ...
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 ...
56 views

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 + ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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) ... 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 ... 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 ... 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 ... 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 ... 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] ... 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, ... 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 ... 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 ? 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 ... 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 ... 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 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)\} ... 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\}. ... 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 ... 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. 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\})$$How can I simplify ... 1answer 35 views ### sum of independent random variables where N is a random variable I want to show E[S_N]=E[N]E[X_j] where: X_1,X_2,\ldots is a sequence of independent random variables, and N is a random variable independent of the sequence. S_n=\sum_{i=1}^n X_i, ... 1answer 50 views ### Solution to Billinglsey (1995) problem 20.22 Let Y_1\leq Y_2\leq ... be random variables s.t. \mathrm{plim} Y_n = Y. Show that Y_n \to Y with probability 1. Some hints? My strategy would be to prove that \sum P(\lvert Y_n -Y \rvert > ... 2answers 34 views ### Heteroskadasticity and Linear Probability Question Suppose (Y,X,U) be a random vector such that$$ Y = X'\beta + U. $$Suppose Y takes values in \{0,1\} and that E[Y\mid X] = X'\beta. Is it reasonable to assume that Var[U\mid X] ... 1answer 88 views ### Upper bound on sum of i.i.d. random variables Here's a problem I've been struggling with: Let X_1, X_2, X_3, \ldots be an i.i.d. sequence of random variables with finite moment generating function M(t). Define the sum S_n = X_1 + \ldots ... 1answer 60 views ### Maximising Entropy of Random Variable taking Positive Integer Values A random variable X takes positive integer values and E[X]=6. What distribution of the random variable X maximises the entropy H(X)? What if X can only take a finite number of values? ... 1answer 100 views ### Exponential distribution: Finding the parameter Please help me solve the following problem Time of production of one electronic component is given with exponential distribution with parameter λ. If the process lasts less than 3 hours, the ... 0answers 57 views ### Copulas/Probability Theory So I have a basic understanding of copulas but wanted to verify I'm applying things correctly to reach the correct outcomes.. Show that as \theta\to\infty, C^{Fr}(u_1,u_2)\to\min(u_1,u_2), the ... 1answer 25 views ### Please help with this probabilities The daily production of a factory is 20 articles, of which two are always defect. A sample of four is taken. Let X be the random variable that assigns the number of defect articles in the sample. ... 1answer 132 views ### Showing certain functions are random variables Assume \{X_k\}_{k \in \mathbb N} are random variables on a probability space. Define induced random walk by S_0 = 0 and S_k = \sum_{i=1}^{k}X_i. Now let n = \inf\{p > 0: S_p > 0\} be ... 1answer 102 views ### show it is a random variable Let X and Y be random variables and let A be an event. Show that the function$$Z(\omega)=\begin{cases}X(\omega) \quad \text{if} \; \omega \in A\\ Y(\omega) \quad \text{if} \; \omega \in A^c ...
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 ...
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 ...