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votes
0answers
46 views

two independent Poisson Arrivals

I have two types of customers (type 1 and type 2) enter a shop. Their arrival processes are independent and follow Poisson process with the arrival rates of $\lambda_1$ and $\lambda_2.$ Consider two ...
0
votes
1answer
167 views

Survival time problem exponential with gamma prior

The survival times, in days, of patients diagnosed with a severe form of a terminal illness are thought to be well modelled by an exponential($\theta$) distribution. We observe the survival times ...
0
votes
1answer
21 views

How to explain the following discrepency when changing parameter for exponential?

Suppose $y$~ $\exp(\frac{1}{2\theta})$ then $\frac1\theta y$~$\exp(\frac12)=\frac14\chi^2_2=\frac14\gamma(1,2)$ then $\sum \frac1\theta y=\frac14\chi^2_{2n}$ then $\sum y$~$ \frac{\theta}{4} \chi^...
1
vote
1answer
288 views

Story Proof relating Poisson and Gamma

I'm having issues proving the following identity: $P(X ≥ j)$ = $P(Y ≤ t)$, where $X$~ Pois($\lambda$$t$) and $Y$ ~ Gamma($j$, $\lambda$) More specifically, I can prove it algebraically but not ...
0
votes
0answers
184 views

Probability of an exponential random variable being greater than a gamma random variable?

Let V have exponential(a) density, and let W be independent of V with gamma(s,b) density. Find P(V>W). What I did for this problem is I integrated the conditional probability P(V>W|W)f(w)dw from w = ...
3
votes
2answers
2k views

Deriving exponential distribution from sum of two squared normal random variables

Let $X$, $Y$ be i.i.d. random variables with distribuition $\mathcal{N}(0,1/2)$ and $Z = X^2 + Y^2$. I'd like to prove based on $X$ and $Y$ pdf's that $Z$ has exponential distribuition.
2
votes
0answers
26 views

Motivation for Gamma, exponential, & chi-squared distributions [duplicate]

I understand the binomial, geometric, Poisson, and normal distributions. However, I don't understand the usefulness or difference between the following: Gamma, exponential, and chi-squared. So far, it ...
3
votes
1answer
163 views

Max n for which sum of exponential distribution is bigger then gamma variable

I am currently preparing to the actuarial exam and it is one of the exercises from previous years I encountered and have no idea how to deal with: Let us assume that $X_1, X_2, ..., X_n$ are ...
5
votes
1answer
587 views

How to choose between exponential and gamma distributions

I have same data and I would like to choose a model for it. To start with I fit an exponential distribution and a gamma distribution. Now I wanted to do a simple likelihood ratio test . However, I ...
0
votes
0answers
27 views

Accounting for minimum dependent measure in data when fitting a distribution

I have what is possible a naive question. I am current comparing various models (i.e. distributions). And the comparisons do not involve different distributions but rather how the model is fed the ...
2
votes
1answer
1k views

Calculating a marginal distribution for the joint density distribution of an exponential distribution with a rate given by a Gamma distribution

This is a follow-up question from one I asked over at MathOverflow: https://mathoverflow.net/questions/158806/is-there-a-simple-closed-form-solution-for-the-joint-density-distribution-of-an The ...
2
votes
2answers
3k views

Manually fitting a mixture distribution in matlab

I am trying to fit a mixture model containing a gamma and an exponential distribution: The general form, using the pdfs, is: p * gammapdf + (1-p) * exponentialpdf. The pdfs for the Gamma and ...
2
votes
1answer
293 views

How can I estimate parameters of a convolution of an exponential and gamma?

Ideally, I would input a one-dimensional array of data, and output the estimates for the three parameters. I'm not very familiar with any statistics software but I have MATLAB (and all of the free ...
3
votes
1answer
965 views

What reference can I cite for the proof that the sum of n exponential variables follows a gamma distribution?

There is a fairly common theorem, which states that: The sum of $n$ independent variables following an exponential distribution $\mathrm{Exp}(\alpha)$ follow an gamma distribution $\mathrm{Gamma} (n, ...
8
votes
1answer
3k views

How to derive Poisson distribution from gamma distribution?

Let $T_1, T_2, \dots$ be iid sequence of exponential random variables with parameter $\lambda$. The sum $S_n = T_1 + T_2 + \dots + T_n$ is a Gamma distribution. Now as I understand the Poisson ...
12
votes
3answers
876 views

How do you calculate the expectation of $\left(\sum_{i=1}^n {X_i} \right)^2$?

If $X_i$ is exponentially distributed $(i=1,...,n)$ with parameter $\lambda$ and $X_i$'s are mutually independent, what is the expectation of $$ \left(\sum_{i=1}^n {X_i} \right)^2$$ in terms of $n$ ...