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1 vote
Accepted

Independence of maximums of independent random iid exponential variables.

Assume $0<m<n$. By symmetry, we have $\mathbb{P}(X_m>X_1,\dots,X_{m-1})=\frac1m$ and $\mathbb{P}(X_n>X_1,\dots,X_{n-1})=\frac1n$ . On the other hand, we have for $0<x<y$ $$(*){:={\...
van der Wolf's user avatar
  • 3,154
1 vote

Simulate a Brownian motion by exponential time stepping

1. The formula in the paper is correct, and here is a more detailed derivation: If $T$ is an exponential random variable with rate $\alpha$, and if $B_t$ is the standard Wiener process independent of $...
Sangchul Lee's user avatar
1 vote
Accepted

How to find the power function given exponential distribution?

You are right. There is an error. The upper limit of integration must be $0.05$. The power is defined by the formula $\inf_{\theta \in \Theta_1} \mathbb{E}_{X \sim \theta}[T(X)]$ for any test $T$. In ...
温泽海's user avatar
  • 2,400
1 vote
Accepted

Integral with two Exponential Distributions

The random variable $$Z = X^{\theta-1} \mathbb 1 (X < Y) = \begin{cases} 0, & X \ge Y \\ X^{\theta-1}, & X < Y \end{cases}$$ has expectation $$\begin{align} \operatorname{E}[Z] &= \...
heropup's user avatar
  • 139k
1 vote
Accepted

Maximum Likelihood Estimation of median for an exponential distribution

This is an exercise in using the invariance property of the MLE. Specifically, if $\hat \lambda$ is the MLE of $\lambda$ for a sample $\boldsymbol X = (X_1, \ldots, X_n)$ where $X_i \sim \...
heropup's user avatar
  • 139k
1 vote

Formal proof of joint pdf for arrival times of a Poisson process

Thanks for clarifying the question in the comments. The intuition you have would be enough if we were talking about probabilities of events. But because we're talking about probability densities, we ...
Ziv's user avatar
  • 131

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