Questions tagged [exponential-distribution]
To be used for questions on using, finding, or otherwise relating to Exponential Distributions.
1,538 questions
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Conditional probability for linear combinations of independent exponentials
I am working on the following exercise.
Let
$$X_1 \sim \mathrm{Exp}\left(\tfrac12\right), \qquad
X_2 \sim \mathrm{Exp}\left(\tfrac12\right),$$
independent. Define
$$Y_1 = X_1 + 2X_2, \qquad Y_2 = 2X_1 ...
- 377
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MGF of standardized $\bar{X}$ when $X_i$ follows exponential distribution [closed]
When $X_1, X_2, ... ,X_n$ follow an exponential distribution, whose $\theta$ is 2,
mgf of $W = \frac{\bar{X}-\mu}{\sigma/\sqrt{n}}$ will be $$\frac{e^{-t\sqrt{n}}}{(1-t/\sqrt{n})^n}$$
I understand the ...
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Density of the sample range $X_{(n)}-X_{(1)}$ and midrange $\frac{X_{(n)}+X_{(1)}}{2}$ of i.i.d. $\exp(\lambda = 1)$ r.v.
I've been trying to find the densities and expected values of the sample range $X_{(n)}-X_{(1)}$ and midrange $\frac{X_{(n)}+X_{(1)}}{2}$ where $X_{(i)}$ denotes the $i$th order statistic from a ...
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naming family of functions defined by average of exponential functions w.r.t. rates
I have a family of functions defined by "a weighted average of exponential functions across rates from any probability distribution", or mathematically:
$$ f(t) = \int e^{-t\lambda} p(\...
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Doubt in Possion and Exponential Distribution co-relation
I was solving the following problem from debore:
At time t = 0, 20 identical components are tested. The lifetime distribution of
each is exponential with parameter λ. The experimenter then leaves the ...
- 104
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Square of Exponential Random Variable
I want to find the pdf of the square of an exponential distribution. Let $X \sim \exp(\lambda)$ and $Y=X^2$. I want $f_Y(y)$.
I start from the $\operatorname{CDF}$ of $Y$: $$ P(Y\leq y)=P(X^2\leq y) \\...
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The power of a test of an exponential distribution
Let $X_1,\dots,X_n$ be a random sample from an exponential distribution with unknown parameter $\theta$. Consider the hypothesis $H_0: \theta=\theta_0$ with alternative $H_a: \theta < \theta_0$, ...
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Sums of exponential random variables with weights geometrically scaled down
We are given independent identically distributed random variables $X_1, X_2, \ldots \sim \text{Exponential}(1),$
and we define $$Y = {\sum_{i=1}^\infty {2^{-i}} \sum_{j=1}^i {X_j}}.$$
The question ...
- 465
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Memoryless Property with light bulbs
Studying properties of the exponential distribution I encountered the following problem:
$n$ light bulbs of different types are lit simultaneously. The lifetimes of these bulbs are independent random ...
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Exponential distribution Problem Cars on Highway
Assume that the distance between cars going in one direction on a certain highway is exponentially distributed with mean value of $100$ meters. What is the probability that in a stretch of $5$ ...
- 1,191
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How to dither binned data (following a geometric distribution) to recover the exponential distribution?
Consider a random variable 𝑋 that follows an exponential distribution. After binning (floor), 𝑋 becomes discrete and follows a geometric distribution. My question is: how can we recover the original ...
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Is the average corner a curve?
Consider the random experiment in which $y_{-1}, y_{+1}$ are i.i.d. exponential random variables with rate parameter $\lambda$, sampled at $x=-1$ and $x=1$, respectively. For each sample, consider the ...
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Why Normal Survival Function decaying slower than the Exponential?
The Normal Survival Function decaying slower than the Exponential on LogLog plot.
Why? Isn't it supposed to be other way around, as Normal is super exponential $\sim e^{-t^2}$ and should decay faster ...
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If sub-exponential distribution condition holds for some $n\ge2$ then it holds for all $n\ge2$
For non-negative random variables $X_1,...X_n\overset{\text{iid}}{\sim} F$, $F$ is said to be a sub-exponential distribution if
$$\lim_{x\rightarrow\infty}\frac{P(X_1+...X_n>x)}{P(X_1>x)}=n$$
...
- 2,396
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Correct answer of $E(W_4| N(1) = 2)$
I'm studying Poisson process and I have questions about how to calculate the expected arrival times conditional on the number of events. I see this exercise in the book "Introduction to ...
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