# Tagged Questions

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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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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 $$\hat{\theta} = ... 2answers 158 views ### Let X_1 and X_2 are independent N(0, \sigma^2) random variables. What is the distribution of X_1^2 + X_2^2? Let X_1 and X_2 are independent N(0, \sigma^2) which means (mean = 0, variance = \sigma^2) random variables. What is the distribution of X_1^2 + X_2^2? My approach is that X_1\sim N(0, ... 2answers 189 views ### Find the pdf of \prod_{i=1}^n X_i, where X_is are independent uniform [0,1] random variables. How do I find the pdf of \prod_{i=1}^n X_i, where X_is are independent uniform [0,1] random variables. I know X~U[0,1], -ln(x) is exponential(1). I also know the sum of two or more independent ... 1answer 43 views ### What is the Var(X-2Y+8)? X,Y are two Random Variables. Var(X)=1\; Var(Y)=2. \rho(X,Y)=\frac16 (\rho=\frac{cov(X,Y)}{\sqrt{Var(X)Var(Y)}}). How I'm calculating Var(X-2Y+8)? I'm using the formula:$$Var\left( ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Derive the Cramer-Rao lower bound (CRLB) for any unbiased estimator of $\mu^2$.

Let $Y_1, Y_2, . . . , Y_n$be a random sample from a normal distribution with mean μ and variance 1. Derive the Cramer-Rao lower bound (CRLB) for any unbiased estimator of $\mu^2$. Could anyone ...
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### Questions related to Rao–Blackwell theorem

In this exercise, we illustrate the direct use of the Rao–Blackwell theorem. Let $Y_1, Y_2, . . . , Y_n$ be independent Bernoulli random variables with $p(y_i | p) = py_i (1 − p)1−y_i , y_i = 0, 1.$ ...
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### find E($\bar{Y^4})$ by using moment generating function for a normal distribution with mean μ and variance 1.

Let $Y_1, Y_2, . . . , Y_n$be a random sample from a normal distribution with mean μ and variance 1. I would like to find E($\bar{Y^4})$ by using moment generating function. The setup I have right ...
### $E[X]< (\sum_{n=0}^\infty P[X>n]< E[X]+1$
If X takes only non-negative integer values then I figured out $$E[X]= (\sum_{n=0}^\infty P[X>n]$$ but I'm having hard time proving $$E[X]⩽ (\sum_{n=0}^\infty P[X>n] ⩽ E[X]+1$$ for any ...