Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.

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2
votes
2answers
59 views

conditional probability about sum and product rule

I am reading Bishop's Pattern Recognition and Machine Learning. In page 73, chapter 2.1. I can't understand the formula 2.19 : $$p(x=1|\mathcal{D})=\int_0^1 p(x=1|\mu)p(\mu|\mathcal{D})\text{d}\mu ...
0
votes
0answers
13 views

Sum of a truncated normal random variable and normal random variable (correlated) [duplicate]

I'm wondering if there is a closed form of pdf of sum of a "correlated" normal random variable and a truncated normal random variable. I found a paper providing the pdf for "uncorrelated case" but I ...
0
votes
2answers
17 views

What is the probability density function of $g(S) =S/2$ for a triangle pdf

Say we have the following "triangle" probability density function: $ p_{S}(s) = \left\{ \begin{array}{lr} s & : s \in[0,1]\\ 2-s & : s \in [1,2]\\ 0 & o.w. ...
2
votes
2answers
92 views

Sum of normally distributed independent random variables, where one has a different (exponential) unit

$$X \sim \mathcal{N}(\mu_X,\,\sigma_X^2)$$ $$Y \sim \mathcal{N}(\mu_Y,\,\sigma_Y^2)$$ $\mu_X$ and $\sigma_X$ have unit decibel watt ($\text{dBW}$); $\mu_Y$ and $\sigma_Y$ have unit watt ($\text{W}$). ...
1
vote
0answers
19 views

Integrating with indicator functions

I want to evaluate $$\int_{-\infty}^{\infty}(A_1e^{-\beta_1(b-x-y)}+B_1e^{-\beta_2(b-x-y)})(pn_1e^{-n_1y}1_{\{y\geq0\}}+qn_2e^{n_2y}1_{\{y<0\}})dy,$$ $b>x, \beta_1<n<\beta_2$. I am trying ...
0
votes
1answer
22 views

Polynomial joint pdf $f(x,y)$ such that of $f(x) \neq f(y)$

How can I build a polynomial joint pdf $f(x,y)$ for $x \in [x_1, x_2]$ and $y \in [y_1, y_2]$ such that of $f(x) \neq f(y)$ or equivalently, $x$ and $y$ are depended on each other?
5
votes
4answers
159 views

What is the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$ and some of the properties of $E[X \mid Y]$?

I was trying to understand both intuitively and rigorously what the difference between $E[X\mid Y]$ vs $E[X\mid Y=y]$. Let me tell you first the things that do make sense to me. $E[X\mid Y=y]$ makes ...
1
vote
1answer
27 views

Probability of multiple variables, geometric distribution?

You are on a basketball team, and at the end of every practice, you shoot half-court shots until you make one. Once you make a shot, you go home. Each half-court shot, independent of all other shots, ...
0
votes
1answer
25 views

How do you get the probability distribution of the sum of random variables by using the inverse of the transform?

I read the following statement: If X and Y are independent random variables, the distribution of their sum W = X + Y can be obtained by computing and then inverting the transform $M_W (s) = ...
0
votes
0answers
33 views

How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable?

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution: $$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$ Is there any ...
0
votes
0answers
24 views

How to estimate the covariance matrix if the unnormalized pdf is known but integral is intractable? [duplicate]

Assume a $d$-dimensional random vector $x$, whose unnormalized pdf is known as the product of N multivariate t-distribution: $$Pr(x)\propto\prod_{i=1}^nt_{\nu_i,\mu_i,\Sigma_i}(x)$$ Is there any ...
0
votes
1answer
36 views

Show that $Pr[X \gg Y]\approx 1$

Can one show (and how) that $$Pr[X \gg Y]\approx 1$$ for $$X:=\sum_{i=1}^k Bin\left(n\left(\frac{1}{2}\right)^i,i\right)$$ and $$Y:=\sum_{i=k+1}^{\infty} ...
0
votes
0answers
63 views

Prove $Pr[X + Y \geq x] \sim Pr[X \geq x]$

We have two independent random variables $X_n$ and $Y_n$, where $$X_n=\sum_{i=0}^n x_i$$ and $$Y_n=\sum_{j=0}^n y_j,$$ where $x_i$,$y_j$ are (non-identically) Bernoulli distributed and independent. ...
1
vote
2answers
37 views

Does $E[X]\gg E[Y]$ for independent RV imply that $Pr[X+Y \geq x] \sim Pr[ X \geq x]$?

We have two independent random variables $X$ and $Y$, where we know that $E[X]\gg E[Y]$, thus $\frac{E[Y]}{E[X]}\rightarrow 0$. I am now interested in $Pr[X+Y \geq x]$ and would like to show that ...
1
vote
1answer
26 views

Singular vector of random Gaussian matrix

Suppose $\Omega$ is a Gaussian matrix with entries distributed i.d.d. according to normal distribution $\mathcal{N}(0,1)$. Let $U \Sigma V^{\mathsf T}$ be its singular value decomposition. What would ...
0
votes
1answer
21 views

Finding the boundaries of integration when calculating P(X + Y > a) or P(X + Y < b) (Jointly Distributed Continuous Random Variables)

I have a problem on setting the boundaries of integration when I'm trying to find probabilities like $P(X + Y > a)$ or $P(X + Y < b)$. For example, when I have $f(x,y) = \frac {x} {5}\ +\frac ...
1
vote
1answer
42 views

Poisson approximation of $X$ by $Poisson(E[X])$

I've tried to find something, but couldn't find anything about the following question. Is it possible to approximate any random variable $X$ with $E[X]=o(1)$ by a Poisson random variable ...
2
votes
3answers
69 views

Poisson distribution given Exponential Distribution

I would need some help on the following problem: We consider two random variables $X$ and $Y$. We suppose that, given X=x, the conditional law of $Y$ is a Poisson distribution of parameter $x$. $X$ ...
0
votes
1answer
45 views

Actuary P/1 Exam Question

I was taking a practice exam and I don't understand a step in one of the solutions to the problem. I understand every part except why $f(y^{0.5})=8(y^{0.5})^{-3}$ If someone could help explain that ...
1
vote
0answers
13 views

Differential Equation Involving a Bivariate PDF and its Marginal CDF

I have a differential equation of the form $$ P(x_1,0) = R(x_1) Q(x_1), $$ where $P$ is an unknown, isotropic bivariate probability density function (pdf) i.e. $$ P(x_1,x_2) = P(x_3,x_4), \quad ...
1
vote
2answers
79 views

Partial sum of binomial

I 'm trying to figure out a closed form solution for the following summation: $\sum_{j=0}^{\omega} j{n \choose j}p^{j}(1-p)^{n-j}$ where $\omega < n$ Is there any closed form solution?
0
votes
2answers
43 views

Probability density function that evolves with time according to a delay differential

Consider a real valued variable $X(t)$ that evolves with time according to the delay differential $\frac{dX(t)}{dt} = \alpha X(t-t_0) \int_{t_0}^\infty f(y) h(t-t_0,y) dy - \beta X(t) ...
3
votes
1answer
119 views

Averaging inverse CDFs

Suppose I have two distributions $P$ and $Q$ on the line that admit well defined inverse cumulative distribution functions $F^{-1}_P$ and $F^{-1}_Q$. I define an "average" distribution $A$ as the ...
1
vote
2answers
34 views

Let $f(x,y)=cx(1-y), 0<x<2y<1$, and $0$ otherwise. Find $c$. Are $x$ and $y$ independent?

I've done a few problems similar to this before, but they have always have something simple like $0 \lt x \lt \infty$ and $0 \lt y\lt \infty$, where choosing the bounds to integrate over is ...
0
votes
1answer
35 views

Probability of Random Variable Minus Random Variable

$X_1 , X_4$ ~ $ Binomial(18000,1/6)$. So $X_1+X_4$ ~ $Binomial(18000,1/3)$. I am asked to find $P(X_1-X_4)\leq 80)=?$. The solution is to find $Var(X_1-X_4)=6000$, $E[X_1-X_4]=0$ and then do the ...
0
votes
1answer
25 views

Finding a pdf from a CDF with a Discrete Random Variable

I know this question isn't very difficult but I'm not convinced I'm doing it right. For a discrete random variable if you have the CDF, the pdf is defined as $f(x)=F(x)-F(x-)$. I have: $$F(x) ...
1
vote
1answer
51 views

Geometric Distribution - How to show that a certain event is unplausible?

We have given a geometric distribution with parameter $p$ as well as some result $r$, which we doubt is an outcome of the given distribution. What is the best way to show that $r$ is indeed not a ...
2
votes
1answer
40 views

Expected Payment under limited policy

The unlimited severity distribution for claim amounts under an auto liability insurance policy is given by the cumulative distribution: $$ F(x) = 1 - 0.8e^{-0.02x}-0.2e^{-0.001x} , x \geq 0$$ ...
1
vote
1answer
60 views

The pdf of $X+Y$

$X,Y$ are independent. $X\sim U(0,1)$ and $$f_Y(y)=\cases{2y,\;0<y<1\\ 0,\;Else.}$$ What is the pdf of $X+Y$? (i.e. $f_{X+Y}$) I know that $$f_X(x)=\cases{1,\;0<x<1\\ 0,\;Else.}$$ But ...
4
votes
1answer
59 views

Expected maximum of a sequence of i.i.d. Poissons

Let $X_i \sim \mathrm{Pois}(1)$ be a sequence of $n$ i.i.d. random variables (with Poisson distribution with parameter 1). I'm interested in the asymptotic behavior of $$\mathbb E[\max_{i \in ...
2
votes
1answer
14 views

Confusion about non-negative mutual information

The formula I was given for calculating information for a specific stimulus $s_x$ is: $$I(R,s_x) = \sum_i p(r_i|s_x) \log_2{p(r_i|s_x)\over p(r_i)} $$ It was also said that information is always ...
0
votes
1answer
38 views

Question about exp. distribution

We know that $X\sim \exp(1),Y\sim \exp(2)$ and they are independent. What is $P(Y>X)$? exp=Exponential... Thank you!
1
vote
2answers
30 views

What is Cumulative Binomial probabilities?

I am new to this so don't know if I am asking the right question as I just read about its usage but didn't know what exactly a Cumulative Binomial probability is. So my question is, What is ...
0
votes
1answer
33 views

Passing thresholds with uniform random variables

I have encountered a challenging task: I have a bunch of uniform random variables "trying" to pass a certain threshold, and another bunch trying to pass a different threshold, and I need to estimate ...
0
votes
0answers
43 views

Is the following probability distribution stationary/constant

For a conservative system, we know that angular momentum, $l$, and total energy, $E$, are constant, i.e. $\dot{l}=\frac{dl}{dt} = 0$ and $\dot{E}=\frac{dE}{dt} = 0$, where $t$ indicates time. Let ...
3
votes
2answers
64 views

$\frac{1}{\sqrt{2\pi}}\int_\frac {1}{2}^0\exp(-x^2/2)dx$

How do we analytically evaluate $J=\frac{1}{\sqrt{2\pi}}\int_\frac {-1}{2}^0\exp(-x^2/2)dx$? This is what I tried: $$ J^2=\frac{1}{{2\pi}}\int_\frac {-1}{2}^0\int_\frac {-1}{2}^0\exp(-(x^2+y^2)/2)dxdy ...
0
votes
0answers
21 views

Frequency distribution, notation problem with the classes

I would like to create a frequency distribution. The range of my data is the interval $[a,b]$. I divide this interval into $n$ equal length part intervals (or classes). So each part interval has the ...
0
votes
1answer
36 views

Understand step in computing marginal distribution of restricted Boltzmann Distribution

Proof taken from http://image.diku.dk/igel/paper/AItRBM-proof.pdf (page 24) I understand everything up to and including: (1) $$p(\textbf{v}) = \frac{1}{Z}e^{\sum_{j=1}^mb_jv_j} \prod_{i=1}^n\sum ...
0
votes
1answer
36 views

Box-Muller method for correlated normals

The standard Box-Muller method produces two independent normal variables given two uniform ones. Is it possible to extend the method such that given a correlation coefficient $\rho\in[-1, 1]$ and two ...
1
vote
1answer
40 views

Find Limiting Distribution of $|X_n|$

Let $Z_1,Z_2,...,Z_n,...$ be a sequence of independent standard normal random variables. Let $X_n=\sum^n_{k=1}\frac{Z_k}{\sqrt{k}}$. Does the limiting distribution of $|X_n|$ exists? If yes, find it; ...
0
votes
1answer
63 views

Operations on Random Variables

It is known that the equivalent resistance of a parallel combination of two resistors is equal to \begin{align*} R = \frac{R_1R_2}{R_1+R_2} \end{align*} which could be also written as ...
-1
votes
2answers
32 views

We are making a Bernoulli experiment… [closed]

We are making series of independent Bernoulli experiment with $\frac13$ chance to success. What is the probability that we got success at the first experiment, if we know that we get two successes at ...
0
votes
1answer
26 views

A question about $\chi^2$ distribution

Ok, i have a question but i start with a definition first so that one can get the context. (All variables in question have the same variance and under $H_0$ which we are considering - they have the ...
0
votes
1answer
37 views

Method of moments for Beta $(\alpha_1,\alpha_2)$ distribution

I am trying to solve for the first two moments of a Beta$(\alpha_1,\alpha_2)$ distribution. We know that the first moment is equal to: $\mu_1 = \frac{\alpha_1}{\alpha_1+\alpha_2}$ and the second ...
3
votes
1answer
58 views

Prove Number of Arrivals $N(s)$ up to time $s$ follows $\mathrm{Poisson}(\lambda s)$ Distribution

This comes from my self-study of Durrett's "Essentials of Stochastic Processes" book, page 97. Definition Let $\tau_1,\tau_2,\ldots$ be independent $\mathrm{exponential}(\lambda)$ random variables. ...
0
votes
1answer
43 views

What are random variables and its connection with functionals?

Here is an image of the conversation which I had with my Prof. (He's the one in violet and myself in orange) The topic was random variables and other probability related definitions. I tried to ...
0
votes
1answer
23 views

Multivariate normal conditioned on sum of squares

Suppose that $X_i$ are i.i.d. N(0,1) random variables, and set $S = \sum_{i=1}^n X_i^2$. Then $S \sim \chi^2_{(n)}$, the $\chi^2$ distribution with $n$ degrees of freedom. Compute the induced ...
0
votes
1answer
13 views

Hypergeometric Distribution Function?

I'm looking for a function that I can use in excel to calculate the probabilities of having certain cards in an opening hand. For example a function that will calculate the probability to get AT ...
1
vote
0answers
40 views

Poisson distribution given Gamma Distribution

I'm struggling with this one: If $\theta $ is a Gamma$(p,\lambda)$ random variable with $p>1$ and $\lambda>0$. We give the density of the gamma distribution: $ f(x) = \frac { { \lambda }^{ p ...
0
votes
0answers
19 views

Dirichlet multinominal Likelihood Derivation

In http://www.gatsby.ucl.ac.uk/~edward/pub/inf.mix.nips.99.pdf, equation 10 to 15 Given $$ p(\pi_1, \pi_2,..., \pi_k) \sim Dirichlet(\alpha/k,\alpha/k,...,\alpha/k) = ...