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### Show that $\frac{X}{X+Y}\sim Beta(\alpha,\beta)$ [duplicate]

Let IG denote Inverse-Gamma distribution Inverse-Gamma. If $X\sim IG(\alpha,1)$ and $Y\sim IG(\beta,1)$. Show that $\frac{X}{X+Y}\sim Beta(\alpha,\beta)$ I tried with jacobian transformation ...
160 views

### How to deduct the coefficient of the Dirichlet distribution? [duplicate]

I am studying the textbook Introduction to Probability Models by Sheldon Ross. In Section 3.6 page 151 (10th edition), the author uses the Dirichlet distribution to deduct the coefficient of a "...
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### how does the beta function make the total probability of Dirichlet distribution integrate to 1? [duplicate]

the Dirichlet distribution $$Dir(\boldsymbol\mu|\boldsymbol\alpha)=\frac{1}{B(\boldsymbol\alpha)}\prod_K\mu_k^{\alpha_k-1}$$ How is the normalization term $B(\boldsymbol\alpha)$ calculated? or in ...
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### Z = X1/(X1+X2) where X1 and X2 are gamma distributed [duplicate]

Suppose that $X_1 \sim \Gamma(\alpha_1,\beta)$ and $X_2 \sim \Gamma(\alpha_2,\beta)$ and let $Z = \frac{X_1}{X_1 + X_2}$ ($X_1$ and $X_2$ are assumed to be independent). I want to prove that $Z$ is ...
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### How can I find the marginal distributions of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$? [duplicate]

Let $X_1 \sim Gamma(\alpha_1,1)$ and $X_2 \sim Gamma(\alpha_2,1)$ be independent random variables. How can I find the marginal distributions of $\frac{X_1}{X_1+X_2}$ and $\frac{X_2}{X_1+X_2}$? By ...
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13 views

### Beta distribution equivalence with two redondant parameters [duplicate]

context In Factor graphs on discrete variables, the parameters are contained in factors associated each with a subset of the random variables in the system. Each factor provides a different positive ...
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9 views

### Two Gamma functions with common terms produces new Gamma and Beta functions [duplicate]

Let X and Y be independent random variables, X ~ Gamma(α,λ) and Y ~ Gamma(β,λ). Prove tha S=X+Y andT =X/(X+Y) are independent ,S~Gamma(α+β,λ) and T ∼Beta(α,β).
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### Product of 2 Uniform random variables is greater than a constant with convolution

I am trying to formulate the following question. X and Y are IID , uniform r.v. with ~U(0,1) What is the probability of P( XY > 0.5) = ? 0.5 is a constant here and can be different. I do respect ...
• 251
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### Marginal distribution of uniform distribution over sphere

Let $(x_1,…,x_n)$ be a random vector uniformly distributed on the $n$-dimensional unit sphere. Is there a closed form solution for the joint distribution of $P(x_1, x_2)$?
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