Linked Questions

-3
votes
3answers
2k views

Find the distribution of the average of exponential random variables [duplicate]

Reading my script from statistics, I have come across the following statement: Consider the following random variables $$X_1,...,X_n$$ which are all independent and identically distributed ...
1
vote
1answer
3k views

How sum of exponential variables is a gamma variable [duplicate]

I have the task to calculate $P(S_{100}\geq 200)$ where $S_{100}=\sum^{100}_{i=1} X_i$ and $X_i$, $i=1,2, \cdots, 100$ are independent $exp(\lambda)$ random variables. One method is to use the fact ...
0
votes
1answer
2k views

Relation between exponential and gamma distribution [duplicate]

If T and S are distributed exponential T,S = Exp(lambda). What is distribution T+S? The solutions says its gamma(2,lambda), but I don't understand why. I can only guess the answer by knowing that E(X+...
1
vote
3answers
513 views

Proof that $\sum_{i=1}^nX_i \sim \operatorname{Gamma}(n)$ [duplicate]

How to prove that $\sum_{i=1}^n X_i$ has a $\operatorname{Gamma}(n)$ distribution, where $X_1,\ldots,X_n$ are independent standard exponentials?
0
votes
1answer
330 views

How to use convolution formula to find distribution of $n$ exponential random variables [duplicate]

Let $W_1, W_2, \dots $ be i.i.d. exponential $(\lambda )$ variables. For each $n$, find the distribution of $T_n = W_1 + W_2 + \cdots + W_n$ I know how to deal with two i.i.d random exponential ...
1
vote
0answers
35 views

Probability density function of sum of random variables [duplicate]

Assume $X_i$ probability density function is : $$f(x,\lambda)=\Bbb{I}_{(0,\infty)}(x)\lambda \exp(-\lambda x)$$ how to find the probability density function of $\sum X_i$ ? The result is $$\Bbb{I}_{(...
4
votes
2answers
2k views

Inverse of a mean, exponential distribution, expected value

Could you help me find the expected value of this random variable? Let $X_1, X_2, ... $ be independent identically exponentially distributed with parameter $\lambda$ random variables. What is the ...
4
votes
1answer
804 views

Waiting time: exponential distribution

Smith is waiting for his two friends Lee and Yang to visit his house. The time until Lee arrives is Exp($\lambda_1$) and the time until Yang arrives is Exp($\lambda_2$). After arrival, Lee stays an ...
0
votes
1answer
693 views

Why is the sample distribution the Exponential distribution Gamma distributed?

Suppose I have $X_i \sim \operatorname{Exp}\left(\beta\right)$. Then, the sample mean of these random variables has gamma distribution (source). $$\overline{X} \sim \operatorname{Gamma}\left( n, \...
1
vote
2answers
298 views

Moment generating function within another function

(a) Let $X$ be an exponential random variable with parameter $\lambda$. Find the moment generating function of $X$. (b) Suppose a continuous random variable $Y$ has moment generating function $M_Y(s)= ...
0
votes
2answers
361 views

CDF of sum of N exponentially distributed random variables with condition

I have $Y=X_1u(X_1-x_{th})+X_2u(X_2-x_{th})+\cdots+X_Nu(X_N-x_{th})$, with all the $X_i\sim\lambda e^{-\lambda}$, $u(t)$ is the unit step function and $x_{th}$ being the threshold which means that any ...
1
vote
1answer
79 views

Probability of sum of random variables: pivotal quantity

I'm trying to find a confidence interval for a parameter $\theta$, yet I'm not quite sure I'm following the right steps. The probability function is such that: $$f(x|\theta) = \frac{1}{\theta} x^{-\...
0
votes
1answer
94 views

A relationship between Poisson distribution and gamma distribution

We define $N(t)$ to be number of events in the interval $[0,t]$. We assume that $N(t) \sim P(\lambda t)$ for $\lambda > 0$. Let $X$ be the waiting time until the $n$-th event, we need to prove that ...
2
votes
0answers
83 views

Distribution of exponential $x_i$s : $\sum_{i=1}^n x_i$

Exercise : Let $X_1, \dots, X_n$ be a random sample from the Exponential Distribution with unknown parameter $\theta$. (i) Find a sufficient and complete statistics function $T$, for $\theta$. ...
0
votes
0answers
80 views

sum of exponentially distributed random numbers

I have a set of exponentially distributed random variables $X_i \sim \exp(\mu_i)$ with rates being also random with some distribution. Is there a way to find the distribution (or the CDF if it is ...

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