# Linked Questions

75 questions linked to/from Generic sum of Gamma random variables
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, ...
• 245
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. ...
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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3 votes
0 answers
303 views

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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. ...
• 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 ...
• 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 ...
• 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 ...
• 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 ...
• 917
13 votes
1 answer
11k views

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}^{-... • 1,463 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 ... • 521 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 ... 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 ... • 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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