0
votes
0answers
21 views

Convergence of Beta Distribution to Bernoulli Distribution [closed]

How will I show that the $$\beta\left(\frac{a}{n} , \frac{b}{n} \right)$$ distribution converges to the $$\operatorname{Bernoulli}\left( \frac{a}{a+b} \right)$$ distribution?
0
votes
1answer
35 views

Question about exp. distribution

We know that $X\sim \exp(1),Y\sim \exp(2)$ and they are independent. What is $P(Y>X)$? exp=Exponential... Thank you!
0
votes
2answers
283 views

Two groups A and B are playing a game…

Two groups A and B are playing a game. The first group that wins 3 times is the winner. The probability that group A will win at on game is $\frac12$ and the same thing for group B. $X$ = The number ...
-1
votes
2answers
29 views

We are making a Bernoulli experiment…

We are making series of independent Bernoulli experiment with $\frac13$ chance to success. What is the probability that we got success at the first experiment, if we know that we get two successes at ...
0
votes
3answers
278 views

What is the probability that A will win…

Two players are rolling two dices, if they get 6 Player A wins, if they get 7, player B wins, else they rolling the two dices again... What is the probability that A will win? I'd like to get any ...
2
votes
1answer
107 views

Deducing an optimal gambling strategy (using martingales).

Apologies in advance for the length, I tried being precise. Suppose a game where in each turn you can gamble a certain amount of money on the result of a fair coin toss. If the coin comes out tails ...
1
vote
1answer
64 views

Exponential Distribution question

I'm having some trouble understanding the mechanics of how to solve with this distribution. The question: The number of years that a washing machine functions is a random variable whose hazard rate ...
2
votes
1answer
54 views

Filling of a tank - recurrence relation

Suppose a tank has a maximum limit of 100 units. Each day 2,1 and 0 units are added to the water level with probability p,r and q. Any excess water would overflow and if it reaches the minimum level ...
1
vote
1answer
56 views

Need help with a basic exercise about Markov chains

Suppose $\left\{ X_{n}\right\} _{n=1}^{\infty}$ is a Markov Chain taking real values. Are the following Markov Chains? $$Y_{n}=\sum_{i=1}^{n}X_{i} , Z_{n}=\left(X_{n},X_{n-1}\right)$$ Edit1 I ...
0
votes
1answer
18 views

Random process with Cauchy distribution

The problem is as follows. Let $X(t)$ be a stochastic process such that $X(t) = V + 2t, t \ge 0$, and $V$ has the Cauchy distribution $x_0 = 0, \gamma = 1$. Find the probability that $X(t) = 0$ for ...
0
votes
0answers
37 views

Probability of Stopping Time Taking specific value - Random Walk 1d

We are considering a simple random walk $(X_n)_{n\in\mathbb{N}}$ starting at $X_0=0$ with $X_n=\sum_{i=1}^nY_i$ where $Y_i$ are iid and $\mathbb{P}(Y_i=1)=\mathbb{P}(Y_i=-1)=\frac{1}{2}$. We want to ...
0
votes
0answers
19 views

The distribution of ratio of two shifted gamma

I am wondering if anyone can help me to find the ratio of this distribution. Assume $S$ and $T$ are independent, where $S\sim Gamma(n-1/2, 4(1+\rho)\sigma^2)$ $S\sim Gamma(n-1/2, ...
0
votes
1answer
19 views

find the marginal probabilty density function

suppose that X and Y are continous random variable with joint pdf given as follows : f(x,y)= { 15y for x^2 <= y <= x } find the marginal probabilty density function?? how can i find marginal of ...
2
votes
0answers
27 views

Geometric ergodicity and mixing - stationary case

I have this theorem: The Markov Chain {$X_n$} is stationary and geometrically ergodic if and only if {$X_n$} is stationary and absolutely regular with $\beta_n=O(\gamma^n)$ for some $\gamma\in ...
1
vote
2answers
19 views

Question about independence

First of all is true that given $X,Y$ two random variables indenpendent; $(X,Y)\in D\subset \mathbb{R}^2$ then $\text{Cov}(X,Y)=0$? I tried to prove it and this is my solution: If $D=[a,b]\times ...
0
votes
0answers
26 views

Unbiased estimator for geometric distribution for $e^\theta$

I want to find unbiased estimator of $e^\theta$ for geometric distribution P(X = k) = $\theta (1-\theta)^{k-1}$. Sample consists of 1 element $X_1$. We can't use maximum likelihood here to find any ...
1
vote
2answers
61 views

Uniformly integrable martingale

I have the following martingale. $M_n=\exp\left(aB_n-\frac{1}{2}a^2n\right)$ for $n\geq0$ and $a\neq0$, $B_n$ is a BM. I have to show that for $a>0$, $M_n\rightarrow0$ in probability. Is $M_n$ ...
0
votes
2answers
74 views

Doob's decomposition of a brownian motion.

Let $B_n$ be a discrete Brownian motion. I need to find the Doob decomposition for ($B_n^2$). Can someone help me please. Thank you in advance.
0
votes
3answers
27 views

Question about $E(|Z|)$ at Normal distribution

$Z$ is a standard normal variable. How do I calculate $E(|Z|)$? ($E(Z)=0$). Thank you!
0
votes
0answers
7 views

Upper 5% value of distribution of the mean

The density function of a random variable x is $f(x)=ke^{-2x^{2}+10x}$. Find the upper 5% point of the distribution of the means of the random sample of size 25 from the above population. I need ...
1
vote
0answers
29 views

A question on recurrent events

In a sequence of Bernoulli trials let E occur when the accumulated number of successes equal to $c$ times the number of failures where $c$ is a positive integer. I need to show that E is persistent if ...
-1
votes
2answers
51 views

find the mean value of x if The probability distribution of a discrete random variable x is given

The probability distribution of a discrete random variable x is $$f (x)= \begin{pmatrix}3 \\ x \end{pmatrix} (1/4)^x (3/4)^{3-x} $$ Find the mean value of x. Construct a cumulative distribution ...
1
vote
2answers
95 views

Let $X\sim N(0,\sigma^2)$. Compute $E(e^X)$ and $Var(e^X)$

I am stuck on the following problem. Let $X\sim N(0,\sigma^2)$. Compute $E(e^X)$ and $Var(e^X)$ Any hints are really appreciated thank you in advance.
0
votes
0answers
15 views

Question on probability distribution theory

Show that for any discrete distribution $\beta_2>=1$. My Attempt: Wikipedia shows $\beta_2=\frac{\operatorname{E}[(X-{\mu})^4]}{(\operatorname{E}[(X-{\mu})^2])^2} {=} \frac{\mu_4}{\sigma^4}$ ...
2
votes
5answers
239 views

Find the distribution of $X_1^2 + X_2^2$? [duplicate]

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$? My approach is that $X_1\sim N(0, ...
2
votes
1answer
27 views

probability theory proof of exponential chebyshev inequality

This is a question about my homework. I am not sure about what is exponential Chebyshev inequality, also how do I get rid of the absolute value and prove it directly by PDF? As well as the ...
1
vote
0answers
35 views

probability theory proof of inequality

proof $Var(X + Y) ≥ 1/2 Var(X) − Var(Y)$ if $X$ and $Y$ are random variable with finite second moment. I believe it has something to do with Markov inequality or Chebyshev inequality, but I don't ...
0
votes
1answer
50 views

Puzzle for Applying the Definition to a t distribution

The coeffcient of variation (CV) for a sample of values $Y_1,\ldots, Y_n$ is defined by $$ CV = S/ \bar{Y}.$$ Let $Y_1,\ldots, Y_n$ be a random sample of size $10$ from a normal distribution with mean ...
1
vote
1answer
138 views

Find the bias for the Maximum-likelihood estimator

Let $X_1,...,X_n$ be a random sample from the pdf $$f(x|\theta) = \theta x^{\theta-1} , 0 \leq x \leq 1, \theta >0.$$ I found the Maximum-likelihood estimator of $\theta$ is $$\hat{\theta} = ...
2
votes
2answers
159 views

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ random variables. What is the distribution of $X_1^2 + X_2^2$?

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$? My approach is that $X_1\sim N(0, ...
1
vote
2answers
193 views

Find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform [0,1] random variables.

How do I find the pdf of $\prod_{i=1}^n X_i$, where $X_is$ are independent uniform [0,1] random variables. I know X~U[0,1], -ln(x) is exponential(1). I also know the sum of two or more independent ...
1
vote
1answer
44 views

What is the $Var(X-2Y+8)$?

$X,Y$ are two Random Variables. $Var(X)=1\; Var(Y)=2$. $\rho(X,Y)=\frac16$ ($\rho=\frac{cov(X,Y)}{\sqrt{Var(X)Var(Y)}}$). How I'm calculating $Var(X-2Y+8)$? I'm using the formula: $$Var\left( ...
0
votes
1answer
34 views

Question about quantile function

Let $X$ be a random variable with cdf $F$. Then the quantile function of $F$ is defined as $$F^{-1}(y)=\inf\{x\in\Bbb{R} : F(x)\ge y\}.$$ Problem. Then there is a random varible $Y$ has a uniform ...
0
votes
0answers
37 views

How to calculate $E(XY)$

If I have a probability function of $X$ and $Y$: $p_{_{XY}}$. We have mug with 3 red balls and 3 white balls. We roll up a dice and we pull out balls as we get at the dice (with no returns). $X$=The ...
1
vote
1answer
84 views

Find $\operatorname{Cov}(\hat{\beta_0}, \hat{\beta_1})$.

Let $Y_1,Y_2,\ldots,Y_n$ and $X_1,X_2,\ldots,X_m$ be random variables with $E(Y_i)=\mu_i$ and $E(X_j)=\xi_j$. Define $$U_1=\sum_{i=1}^n a_i Y_i\quad\text{and}\quad U_2=\sum_{j=1}^m b_j X_j$$ for ...
1
vote
1answer
48 views

Show that the least squares estimator of the slope is an unbiased estimator of the `true' slope in the model.

Under the assumptions of the classical simple linear regression model, show that the least squares estimator of the slope is an unbiased estimator of the `true' slope in the model. Anyone have any ...
0
votes
1answer
24 views

Poisson Distribution expected value problem

The number of cars (X) arriving at a service station per day follows a Poisson distribution with mean 4. The service station can provide service to a maximum of 4 cars per day. Then the expected ...
2
votes
0answers
40 views

A right-inverse of Brownian motion local time at zero has stationary independent increments

Let $L_0^t$ be the local time for a standard Brownian motion at $0$ and define $$X_t=\sup\{s\ge0:L_0^s\le t\}, t\ge0. $$ I would like to show that $(X_t)$ has stationary independent increments. That ...
0
votes
1answer
70 views

Random Variable depending on another Random Variable?

Here is a quesiton that I was also able to found on the Internet, here. Actually, I've solved 4 of 5 questions, so I only show you the question that I could not. A Computer system carries out ...
0
votes
1answer
50 views

Find unbiased for $\theta$ of $\hat{\theta_1} = (n+1) ~y_{(1)}$given a uniform distribution on the interval $[0, \theta]$.

Show that $\hat{\theta_1} = (n+1) ~y_{(1)}$ is unbiased for $\theta$. For $$P[Y_i \le Y] = 1 - y/\theta$$ Then for $ P[Y_{(1)} < y] = 1 - [1-F_{(Y_i)} (y)]^n$ which should equal to $1 - ...
0
votes
1answer
23 views

Use $2\sum_{i=1}^n Y_i/\beta$ which is a pivotal quantity to derive a 95% confidence interval for $\beta$

Suppose $Y_1$,...$Y_n$ is a random sample from a gamma distribution with $\alpha = 2$ and unknown $\beta$. GOAL: Use $2\sum_{i=1}^n Y_i/\beta$ which is a pivotal quantity to derive a 95% confidence ...
0
votes
0answers
20 views

Basic Questions for MGF and Pivotal Quantity

Suppose $Y_1$,...$Y_n$ is a random sample from a gamma distribution with $\alpha = 2$ and unknown $\beta$. I have used the moment generating function to find that $$m_u (t) = E(e^{tu}) = ...
0
votes
1answer
25 views

Fun with probability measure functions

We have a transition matrix $M =(p_{i,j})$, i and j being elements of the set S, and a probability measure function $f_0(i_0)$, on the set S. Define for all natural numbers n $f_n(i_0,...,i_n)= ...
0
votes
1answer
25 views

Calculating independence of two random variables

I have two random variables $X$ and $Y$. $X$ takes the values ${-1,0,1}$ and $Y = X^2$. I have to determine if these two are independent. I have already calculated that the covariance = 0 for these ...
2
votes
1answer
58 views

How to use the Cramer-Rao lower bound (CRLB)to show that $\bar{Y}$ is the best unbiased estimator of $\lambda$?

Let $Y_1,\ldots,Y_n$ be a random sample from Poisson ($\lambda$). Derive the Cramer-Rao lower bound (CRLB) for the variance of any unbiased for estimator of $\lambda$. MY APPROACH: $$ ...
1
vote
2answers
92 views

Show that $S = \sqrt{S^2}$ is a biased estimator of $\sigma$ given a random sample from a normal distribution …

Suppose $Y_1, \ldots, Y_n$ is a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Let $S^2$ be the sample variance, which is unbiased for $\sigma^2$. GOAL: Show that ...
1
vote
1answer
37 views

Show $E(\frac{1}{2\theta\sum_{i = 1}^n W_i}) = \frac{1}{2(n-1)}$ given the pdf $f(y\mid\theta) = \theta y^{\theta - 1}, 0 < y < 1, \theta > 0 .$

Let $Y_1, \ldots, Y_n$ be a random sample from the pdf $$f(y\mid\theta) = \theta y^{\theta - 1}, 0 < y < 1, \theta > 0 .$$ show that $$E\left(\frac{1}{2\theta\sum_{i = 1}^n W_i}\right) = ...
0
votes
1answer
83 views

Derive the Cramer-Rao lower bound (CRLB) for the variance of any unbiased for estimator of $\lambda$ for random sample from Poisson ($\lambda$)

Let $Y_1,...,Y_n$ be a random sample from Poisson ($\lambda$). Derive the Cramer-Rao lower bound (CRLB) for the variance of any unbiased for estimator of $\lambda$. I first set $$L(\lambda) = ...
0
votes
1answer
34 views

Use the Maximum Likelihood Estimation approach to find an estimator for $\alpha.$ given the Pareto distribution

Let $Y_1,...Y_n$ be a random sample from the Pareto distribution with parameters $\alpha$ and $\beta$, where $\beta$ is known. Then, if $\alpha > 0$, $$f(y|\alpha, \beta) = \alpha \beta^\alpha ...
1
vote
1answer
86 views

Given the pdf $f(y\mid\theta) = \theta y^{\theta - 1}, 0 < y < 1, \theta > 0 .$ show $E(\frac{1}{2\theta\sum_{i = 1}^n W_i}) = \frac{1}{2(n-1)}$

Let $Y_1, \ldots, Y_n$ be a random sample from the pdf $$f(y\mid\theta) = \theta y^{\theta - 1}, 0 < y < 1, \theta > 0 .$$ show that $$E\left(\frac{1}{2\theta\sum_{i = 1}^n W_i}\right) = ...