1
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2answers
50 views

Expected values with exponentials

I've been stuck on this question for a while and it's annoying the hell out of me! I know it's a basic definition type of question, but I can't seem to understand it. Can any of you help? Question: ...
1
vote
1answer
20 views

Order of growth in uniform distribution

Consider an i.i.d. sample $\{X_1, \ldots , X_n\}$ from the uniform distribution on $[ 0,\theta]$ and the estimator $$M_n = \max\{X_1,X_2,\ldots,X_n\} $$ What does the above statement mean? I ...
3
votes
1answer
62 views

Details from a Proof that a Tournament has Property $S_k$

(Edit: While the context is not central to my question, I decided to include it anyway to make the question a little more searchable.) Some technical details are omitted from a theorem in Alon and ...
0
votes
0answers
25 views

Overflow and underflow of a probability value

I am evaluating the probability that the minimum of a process is a above a a barrier $\log(H)$. The probability is given by $$P_i=1-\exp\left(-2\frac{(\log(H)-x)(\log(H)-x_b)}{\tau\sigma^2}\right).$$ ...
2
votes
1answer
74 views

Proving $E_{\theta}[T(X)] = \frac{\psi'(\theta)}{\eta'(\theta)}$

I'm trying to understand how to prove the following theorem: Let $\{P_{\theta}, \theta \in \Theta\}$ be a family of distributions in the one parameter exponential family with density (pmf) ...
1
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1answer
67 views

independent Exponential distribution P(X > Y + 1)

$X$ and $Y$ are independent exponentially distributed random variables with parameters $a$ and $b$. Calculate $P(X > Y + 1)$. I have let $X-Y=Z$ and Then $P(Z>z)=1-P(Z\leq z)$ $1 - P(X-Y\leq ...
0
votes
2answers
46 views

Probability Random Variable question Need Help Please

You have a set of ten light bulbs - the lifetime of each of them being given by an exponential RV with mean 1000 hrs. Find the probability that.... (a) at least 7 of the bulbs function for 1500 or ...
1
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1answer
48 views

$\mathrm{Ei}(x)$, the exponential function, some question.

I have a question involving with $\mathrm{Ei}(x)$, define as $\int_{-x}^{\infty}e^u \cdot u^{-1} \mathrm{d}u$. My question is, when I have a expression say $\exp(x) \cdot \mathrm{Ei}(x)+1$. I want ...
0
votes
1answer
30 views

Exponential Distribution as a density function

I have an important presentation on tuesday about the exponential distribuion as a density function. My question is: What are the advantages of using this function? In order to fulfill my task i have ...
2
votes
2answers
61 views

What's the density of $Z=\max(X,Y)-\min(X,Y)$ with $X,Y$ exponentials of parameter $\lambda$?

Let be $X,Y$ two independent exponential random variables with parameter $\lambda$. What is the pdf of $Z=\max(X,Y)-\min(X,Y)$? Thanks for your help.
0
votes
1answer
23 views

Exponential Random Variables and Confidence

Assume that the amount of evidence against a defendant in a criminal trial is an exponential random variable X. If the defendant is innocent, then X has mean 1, and if the defendant is guilty, then X ...
1
vote
1answer
51 views

Confusion about an algorithm making a choice between two options, with probabilities.

I am totally puzzled at grasping the meaning of "we move to B with probability P1 OR we move to C with probability P2" in the following scenario. A,B,C are points in a 64-dimensional space. Reading ...
0
votes
0answers
17 views

Scaling model output to be between 0 and 1

I have fitted Cox model and the output is generated as: $e^{\beta x}$, where $\beta$ is the coefficient. Now, I would like to have the model output ranging between $0$ and $1$. I'm currently using ...
1
vote
1answer
57 views

What's the MLE of lambda for $f(x)= \frac{1}{λ}\exp{\frac{−x}{λ}}$?

What's the MLE of $\lambda$ for $$f(x)= \frac{1}{λ}\exp\left({\frac{−x}{λ}}\right)$$ Values of x are 5,7,9,3,6,8 Is it just the mean of $x$? Thanks.
1
vote
1answer
62 views

Product of exponential distributions

Suppose $X_1$ is $\mathrm{Exp}(\lambda_1)$ and $X_2$ is $\mathrm{Exp}(\lambda_2)$. $X_1$ and $X_2$ are independent. Let $Y = \min (X_1, X_2)$ and $Z = \max (X_1, X_2)$ and $W = ZY$ . Compute the ...
2
votes
1answer
28 views

n events of one process occuring before m events of another process

Assume that you have two independent Poisson processes, N1( t ) with rate λ1 and N2( t ) with rate λ2 . What is the probability that n events occur in the first process before m events occur in the ...
0
votes
1answer
39 views

Lifetime of exponential variable of a battery

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with parameter $\theta=2$ $($measured in years$)$. Find the probability that a battery of this type ...
0
votes
1answer
36 views

Expected Value of Exponential

I want to calculate $\log E[\exp(-\sqrt{d} S \epsilon)]$, where $\epsilon \sim N(0,1)$ and everything else is deterministic. The result should be $\frac{d}{2}||S||^2$ but why?
1
vote
0answers
21 views

The variance of an arrival process with shifted exponential interval

Here we have a arrival process. The inter-arrival time follows a shifted negative exponential distribution as: $f(t)=e^{−\lambda(t−\theta)}$ How to derive the variance of the number of arrivals in ...
0
votes
1answer
142 views

probability of maximum of two independent random variable

Suppose $X$ and $Y$ are two independant random variable with exponential distribution with paramet $\lambda=1$ and $M=$max{$X$,$Y$}. Then $P(M \ge 4)$ is equal to : Answer: 0.036 how do i come to ...
2
votes
1answer
68 views

How to prove that if $\int_t^\infty(s-t-\frac{1}{\lambda})\,f(s)\ ds =0$ for all $t\ge 0$ then $f(s)=\lambda\, \mathrm{e}^{-\lambda s}$

The problem is motivated by my probability text which states that if the expectation of time to wait conditioned on time already spent waiting $(t)$ is constant (equals $\frac{1}{\lambda}$) then the ...
1
vote
2answers
227 views

PDF and CDF of the division of two Random variables

I have two RVs; their PDF are as the followings: \begin{split} f_{X}(x) = \frac 1 {a} e^{-\frac x {a}}\end{split} and \begin{split} f_{Y}(y) = \frac {y^{L-1}} {b^{L} \Gamma (L)} e^{-\frac y ...
3
votes
1answer
389 views

The Exponential decay.

I am studying semiconductor physics. there is a paragraph about Drude model in E.spenke's book "Electronic semiconductors" page 259 in art §9: "if on the average, a time $τ$ elapses between two ...
1
vote
0answers
61 views

Negative exponential/ exponential power distribution between 0 .0 and 1.0?

Note: I'm not very familiar with distribution and higher level math Heyho, I'm currently looking for a way to generate random values between 0.0 and 1.0 with an exponential power or negative ...
2
votes
1answer
56 views

Random range that iteratively multiplied by itself tends to 1

I was trying to emulate a random currency value, and first thing I thought was to start with value 1 and then iteratively add, say, Rand(-.1, .1), which is a ...
2
votes
0answers
64 views

Application of exponential distributions

The magnitudes of earthquakes in a region of North America can be modeled by an exponential distribution with mean 2.5 (measured on the Richter scale). If 3 earthquakes occur in a given month, what is ...
1
vote
1answer
27 views

Joint distribution proof

I am trying to study for an exam and I am kind of lost on how my professor came to a particular result on his practice exam. Let $W$ be an exponentially distributed random variable with $\lambda = 2$ ...
0
votes
1answer
33 views

Is the distribution of one exponential will be smaller than a second one Uniform?

I came by an expression which I am not sure I understand. If: $X_1 \sim exp(\lambda)$ $X_2 \sim exp(\lambda)$ Then: $P(X_1<X_2|X_2) \sim Uniform(0,1)$ Where it is not clear to me what ...
1
vote
1answer
62 views

Calculating cumulative Markov Chain outcomes

I have a Markov process, with 2 possible states (1 or 0) and a transition matrix P. State at time t=n is determined by x0*Pn. As n goes to infinity, xn goes to the steady state vector, q = [q1 q2]. ...
1
vote
1answer
110 views

How to vary lambda in exponentially distributed numbers

I am implementing an exponentially distributed random number generator (RNG) based on George Marsaglia's Ziggurat algorithm. I previously used the algorithm to create a normally distributed RNG. By ...
1
vote
0answers
41 views

Inequality of Partial Taylor Series

For a given $\theta < 1$, and $N$ a positive integer, I am trying to find an $x > 0$ (preferably the smallest such $x$) such that the following inequality holds: $$\sum_{k=0}^{N} \frac{x^k}{k!} ...
1
vote
1answer
239 views

Stuck on solving for x in exponential to find variance

The problem seems simple: Let X be an exponential random variable such that $P(X \le 2) = 2P(X > 4)$. Find the variance of X. Easy, right? $ P(x \le 2) = 1 - e^{-2\lambda} $ and $ P(x > 4) = ...
3
votes
4answers
115 views

$X$ is $\text{Exp}(\lambda)$, $Y$ is $\text{Uniform}(0,X)$. How can I find $\Bbb E[Y]$ and $\text{Var}(Y)$?

$X$ is $\text{Exp}(\lambda)$, $Y$ is $\text{Uniform}(0,X)$. How can I find $\Bbb E[Y]$ and $\text{Var}(Y)$? I did tried to plug it like double integral of $\Bbb E[Y]$ from 0 to X which $f(t)$ is ...
3
votes
1answer
219 views

exponential population growth models using $e$?

Im trying to understand this write up [1] of cell population growth models and am confused about the use of natural logarithms. If cells double at a constant rate starting from 1 cell, then their cell ...
2
votes
0answers
60 views

differential operator

I've read journal "On the Comparison of Several Mean Values: An Alternative approach" (Welch, 1951). I don't understand this expression: $$E\left(\exp\left[ \sum_t ( w_t - \omega_t ) ...
2
votes
1answer
679 views

Conditional Expected value for exponential distribution

I meet a problem when solving a exponential distribution problem. The problem is to calculate a conditional expectation value for two independent exponential distribution with rate ${\mu _1},{\mu ...
0
votes
1answer
660 views

X1 X2 independent variables exponential distribution - Looking for simpler solution

Let $X_1 \sim \exp(\lambda)$ and $X_2 \sim \exp(\lambda)$ be two independent exponentially distributed random variables. Find the mean and variance of random variable $Y=X_1 + X_2$. $x=x_1 + ...