Linked Questions

10 votes
1 answer
5k views

The distribution of the linear combination of Gamma random variables [duplicate]

If $X_i\sim\Gamma(\alpha_i,\beta_i)$ for $1\leq i\leq n$, let $Y = \sum_{i=1}^n c_iX_i$ where $c_i$ are positive real numbers. Assume all the parameters $\alpha_i$'s and $\beta_i$'s are all known, ...
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0 votes
1 answer
2k views

PDF and CDF of sum of two independent $\Gamma$-distributed random variables [duplicate]

Let $X \sim \Gamma(m, p)$ with a shape parameter $m$ and a scale parameter $p$ and $Y \sim \Gamma(m, q)$ with a shape parameter $m$ and a scale parameter $q$, and let $X$ and $Y$ be independent. ...
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5 votes
0 answers
287 views

Asymptotic distribution of a weighted sum of chi squared variables beyond CLT? [duplicate]

I have a sum $$ S = \sum_{i=1}^{n} d_i X_i^2, $$ where $X_i$ are independent standard normals, and $d_i > 0$ are fixed real numbers, for example $d_i = i$. The asymptotic distribution of this sum ...
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  • 537
3 votes
0 answers
303 views

Quantiles of linear combination of independent $\chi^2_1$ random variables [duplicate]

I want to work out the quantiles of a linear combiation of chi square random variables. Suppose $\lambda_i \in \mathbb{R}$ for all $i \in \{1,2,\cdots,n\}$ and $Z = \sum_{i = 1}^n \lambda_iX_i$ and $...
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0 votes
0 answers
33 views

Is this distribution Chi-Squared? [duplicate]

Let's assume $Y=\sum_{i=1}^{N}\alpha_iX_i^2$. where $X_i$ has Gaussian distribution with mean $0$ and variance $1$, i.e., $\mathcal{N}(0,1)$ and $\alpha_i$s are constants. When $\alpha_i$s are all $...
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  • 21
0 votes
0 answers
15 views

what is the simplified expression for the sum of probability of exponential distributed data? [duplicate]

Hi, I want to find the expression of P(Y1+...+Yn>U) as shown in the screeshot below but I have no idea how to do this. I'm not sure whether I should use Erlang distribution or gamma distribution. ...
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  • 113
55 votes
10 answers
58k views

Why is the sum of two random variables a convolution?

For long time I did not understand why the "sum" of two random variables is their convolution, whereas a mixture density function sum of $f(x)$ and $g(x)$ is $p\,f(x)+(1-p)g(x)$; the arithmetic sum ...
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  • 11.9k
50 votes
3 answers
10k views

How does saddlepoint approximation work?

How does saddlepoint approximation work? What sort of problem is it good for? (Feel free to use a particular example or examples by way of illustration) Are there any drawbacks, difficulties, things ...
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  • 261k
42 votes
7 answers
20k views

How often do you have to roll a 6-sided die to obtain every number at least once?

I've just played a game with my kids that basically boils down to: whoever rolls every number at least once on a 6-sided die wins. I won, eventually, and the others finished 1-2 turns later. Now I'm ...
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  • 1,588
8 votes
1 answer
17k views

Distribution of sum of squares of normals that have mean zero but not variance one?

I am trying to find the distribution of a random variable that is calculated according to $Y:=\sum_{i=1}^n X_i^2$ where $X_i $ is distributed as $ \mathcal{N}(0,\sigma^2_i)$. Does there exist a ...
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  • 917
13 votes
1 answer
11k views

Distribution of sum of exponentials

Let $X_1$ and $X_2$ be independent and identically distributed exponential random variables with rate $\lambda$. Let $S_2 = X_1 + X_2$. Q: Show that $S_2$ has PDF $f_{S_2}(x) = \lambda^2 x \text{e}^{-...
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9 votes
2 answers
8k views

Real life uses of Moment generating functions

In most basic probability theory courses your told moment generating functions (m.g.f) are useful for calculating the moments of a random variable. In particular the expectation and variance. Now in ...
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10 votes
4 answers
4k views

Expected number of dice rolls require to make a sum greater than or equal to K?

A 6 sided die is rolled iteratively. What is the expected number of rolls required to make a sum greater than or equal to K? Before Edit ...
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2 votes
1 answer
16k views

Does the sum of two independent exponentially distributed random variables with different rate parameters follow a gamma distribution?

Suppose we have two independent exponentially distributed random variables with means $400$ and $200$, so that their respective rate parameters are $1/400$ and $1/200$. Do these random variables then ...
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  • 149
8 votes
1 answer
4k views

Generalized Chi-Squared Distribution PDF

Let $\mathbf{X} \sim \mathcal{N}_n( \mathbf{m}, \mathbf{C})$ be an $n$-dimensional gaussian vector, where $\mathbf{C} \in \mathbb{R}^{n \times n}$ is not diagonal, but it is positive-definitive, $\...
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