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

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0
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1answer
25 views

Given the moment generating function of a continuous-type r.v, how to find the p.d.f?

Say for $t<1$: $$M(t) = \frac{1}{(1-t)^2}$$ How to find the p.d.f of the random variable? $$M(t) = E(e^{tx})=\int_{-\infty}^{+\infty}e^{tx}f(x)dx$$ How do we find: $f(x) = xe^{-x}$ on ...
2
votes
3answers
43 views

Distribution of sum of random variables

Let $X_1, X_2, . . .$ be independent exponential random variables with mean $1/\mu$ and let $N$ be a discrete random variable with $P(N = k) = (1 − p)p^{k-1}$ for $k = 1, 2, . . . $ where $0 ≤ p < ...
0
votes
2answers
38 views

Neighbor Interaction in a Random List

Assuming a random arbitrarily long list where each element has a $50\%$ chance of being a $0$ or a $1$, such as: $0001101101$ What is the chance of having a neighbor that isn't the same? For ...
0
votes
1answer
33 views

Find the probability density function of $Y=X+Y$

If the joint density of $X_1$ and $X_2$ is $$f(x_1,x_2) = 6e^{-3x_1-2x_2}, x_1 >0; x_2>0$$ Find the probability density function of $Y=X_1+X_2$ We did this as a class example but never ...
0
votes
1answer
26 views

Can the independence of random variables hold for their functions?

Suppose $X$ and $Y$ are two independent continuous random variables on $\mathbb{R}$. Define: $f:\mathbb{R}\mapsto\mathbb{R}$ as a $C^\infty$ map on $\mathbb{R}$. Then is it possible to find the ...
0
votes
0answers
12 views

Approximate CDF of arbitrarily aggregated random variable

I would like to know if my solution for the following is mathematically correct in general: I have a random variable $Z$ that is an arbitrary function of two other rvs $X$ and $Y$, so: $Z = f_{arb}(X, ...
0
votes
2answers
22 views

There's something fundamental that I'm missing regarding the Standard Uniform Distribution (Continuous)

So if we have the standard uniform distribution $$X \sim U(0,1)$$ $$f(x) = 1 \text, 0 < x < 1$$ So now I don't understand how the probability of having any point between 0 and 1 is just 1. ...
0
votes
1answer
14 views

Variance of Transformed Random Vectors

Consider an $n$-dimensional normal random vector $\mathbf X:= (X_1, \dots, X_n)^T$ with mean $\mathbf 0$ and covariance matrix $\mathbf \Sigma$. Now define a new random vector $\mathbf Y:= (a_1X_1, ...
0
votes
1answer
34 views

Finding expected number of trials until we get head given density function?

Suppose we flip a coin with a random probability of Heads $P$ that has density $f(p) = 6p(1−p),\; p \in [0, 1]$. If we keep on flipping this coin until we get a single Heads, what is the expected ...
4
votes
1answer
78 views

How long before the prey can escape?

I've (sort of) come across the following problem in my research. The actual scenario is a little abstract to explain, so I'm rephrasing the problem in terms of a predator/prey scenario. I'm tagging ...
2
votes
1answer
67 views

How to find the density of $Y=g(X)$ in this case?

I have a vector $X=(1,X_2,X_3)$, where $(X_2,X_3)$ is a random vector in $\mathbb{R}^2$. Now consider $Y=g(X)=X/\|X\|$. What is a density function of $Y$ with respect to the uniform spherical ...
-1
votes
0answers
17 views

p.d.f of a random variable $X$ [duplicate]

Let $f(x) = x/15$ if $x \in \{1,...,5\}$ and $0$ elsewhere be the p.d.f of the random variable $X$. Is $f$ considered of continuous or discrete type? Why? And what is the difference between: $P(X ...
9
votes
1answer
127 views

Math Intuition and Natural Motivation Behind t-Student Distribution

I am trying to understand with basic mathematical background how the $t$-Student distribution is a "natural" $pdf$ to define. So I hope that this not too-general a question, but given that the ...
3
votes
1answer
30 views

Cancellation law of equal in distribution

I came across this gem while discussing with my friends, If $X$ and $Y$ are two real valued random variables (not necessarily independent) that satisfy $$X =^d X+Y$$ (where $=^d$ means equal in ...
1
vote
0answers
28 views

Proof that Sum of $n$ Squared Errors ~ Chi Square with $n$ $df$

There is a youtube video dealing with the proof that the sums of the squares of normally distributed $n$ random errors, each one distributed as $\sim \chi^2(1\text{ df})$ follows a chi square ...
0
votes
2answers
37 views

Get the distribution of $X|Y=y$ given this joint probability density function

Given the joint probability density function $f(x,y) = \lambda^2 \exp(-\lambda y)$ with $0 < x < y.$ How do I get the distribution of $X|Y=y$ ? Thanks in advance!
4
votes
2answers
152 views

Help with C is Euler's constant and $\Gamma(0)=\infty$ in paper

I am referring to a paper by S. Nadarajah & S. Kotz. The notation is simple enough to understand, however i having trouble with $C$ is Euler’s constant and $\Gamma(0)=\infty$ by equation (2.3) I ...
0
votes
1answer
56 views

Find the CDF of a function of two random variables

The joint probability density function of two continuous random variables $X$ and $Y$ is: $$f(x,y) = \begin{cases} 6x,& 0\leqslant x\leqslant y,\ 0\leqslant y\leqslant 1\\ 0,& \text{ ...
2
votes
1answer
29 views

Degree of Polynomial in Centered Moments of Gamma$(n,1)$

I'm interested in the degree of the polynomial in $n$ of the expression for the $k$-th central moment $$ E((X_n - n)^k) $$ where $X_n$ is a Gamma$(n,1)$ random variable, that is, the sum of $n$ ...
1
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2answers
28 views

Weighted sum of identical distributed random variables

Suppose $X_1$, $X_2$, $\ldots$ ,$X_N$ are identically distributed (not necessarily independent). Then, given $a_1+a_2+\ldots+a_N=1$, and let $S=a_1 X_1 + a_2 X_2 + \ldots + a_N X_N$. Does $S$ follow ...
-1
votes
1answer
54 views

Cards probability problem

Two players; the dealer and a player. The player is given three cards face down. The dealer turns over a 2 (let's say of hearts). Before the player turns any cards over, what is the probability that ...
1
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1answer
47 views

Probability and Name of Distribution

Suppose $X$ has PDF $f_X$ given by \begin{align*} f_X (x) = \begin{cases} \frac{\alpha x_0^\alpha} {x^{\alpha+1}} &\text{if $x ≥ x_0$,}\\ 0 &\text{if $x < x_0$,}\end{cases} \end{align*} ...
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0answers
34 views

Transition density of an AR(1) process?

If we have an AR(1) process, i.e: $X_{t+1} = \alpha X_t + e_{t+1}$ with $X_0=0$ then what is its Markov Chain transition density? We know that for a Markov chain, the following holds: $P(X_{t+1}\leq ...
0
votes
1answer
38 views

Moment generating functions…which distributions to use?

Q: You hired a terrible programmer and the moment generating function for the distribution of software bugs is M(t) = (1 - $\theta$t)$^{-\alpha}$. Groups of bugs can be detected within $\mu$ = 47 ...
0
votes
1answer
29 views

Find conditional probability of a mixture model

given is the following: A mixture model comprises a non-observable $\{ 0,1\}$-valued random variable $X$ such that $P(X=1)=1-P(X=0)=\pi$ and an observable variable $Y$ such that $Y\mid X=0$ is ...
1
vote
1answer
20 views

Linear regression relationships

Velocity $= X$, distance to stop $= Y$ $\beta_0= -17.5791$, $\hat{\operatorname{se}}(\beta_0)=6.7584$ $\beta_1 = 3.9324$, $\hat{\operatorname{se}}\beta_1 = 0.41.55$ degrees of freedom $=48$ (a) is ...
0
votes
1answer
17 views

What am I plugging in wrong to my normal distribution calculator?

I am trying to find the probability of the following question: Cans of regular Coke are labeled as containing 12 oz. Statistics students weighed the contents of 7 randomly chosen cans, and found the ...
1
vote
1answer
27 views

Distribution Technique Question of two independent Exponential Distributions

If $X_1$ and $X_2$ are two independent random variables having exponential densities then $f(x_1,x_2)$ is defined as $$f(x_1,x_2)=\exp(-(x_1+x_2))\,{\bf 1}_{(0,\infty)}(x_1){\bf ...
0
votes
0answers
28 views

Weighting the data by the history

I have a input stream 3D data that comes every time frame. Each point is defined by 3D vector of x,y,z. There is a evaluation function [say f(x)] that computes if the point at time t is valid or ...
1
vote
1answer
55 views

Runs of white balls in sampling without replacement

There are $m$ white balls and $n$ black balls in a box. Balls are randomly drawn from the box with no return. Denote $X_1$ : number of white balls that been drawn before the first black. For $2 \leq i ...
-1
votes
1answer
32 views

Normal distribution calculations

We have a gaussian distribution $$ X \sim N(\mu,\sigma^2)$$ where $\mu = 4$ and $\sigma^2 =1.5$ . Probability is given by : $P(x<c)=0.35$ $c$ needs to be calculated. And we got ...
1
vote
2answers
39 views

Derivation of mean and variance of Hypergeometric Distribution

I need clarified and detailed derivation of mean and variance of a hyper-geometric distribution. If a box contains $N$ balls, $a$ of them are black and $N-a$ are white, and $n$ number of balls are ...
6
votes
1answer
82 views

Multivariate normal density function of function of random variable

Let $X_1,\dots,X_n$ be i.i.d random variables and $g$ be a symmetric function such that $$g(X_i,X_j)\sim N(\mu,\sigma^2)$$ for all $1\le i<j\le n$. I wish to know the density function of the joint ...
-1
votes
1answer
16 views

Distribution of specific distributions

I have a normal distribution of independent variables, and there are a specific number of samples to this distribution: say 1 million samples. A function is set by the largest value of these million ...
0
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1answer
33 views

Seeking an example for Bayes estimator of two unknown parameters

I searched the web, taking advantage of several search approaches; however, due to redundancy of the existing information about Bayes estimator of one unknown parameter of random variables (either in ...
0
votes
1answer
18 views

If X + Y is truncated normal and X and Y are identitically (but not independently) distributed? What is the distribution of X and Y?

Let $(aX + bY)$ be a truncated normal and assume $X,Y$ are both identically distributed (but necessarily NOT independent) what is the distribution of $X$ and $Y$? More importantly can the pdf of $X$ ...
2
votes
2answers
69 views

Probability distribution of number of waiting customers in front of a counter [closed]

The number of customers arriving at a bank counter is in accordance with a Poisson distribution with mean rate of 5 customers in 3 minutes. Service time at the counter follows exponential distribution ...
4
votes
2answers
74 views

Lower bound for (function of) density of well-behaved random variable

Suppose we have a non-negative random variable $\tilde{\theta}$ such that $\mathbb{E}\tilde{\theta} = a > 0$, with finite variance $\sigma^2$. $\tilde{\theta}$ can take on values from $0$ to ...
4
votes
1answer
34 views

Flat product distribution

When $X,Y$ are iid random variables, uniformly distributed on $[0,1]$, then $Z=X*Y$ has the density $-\log(z)$, as it is shown here: product distribution of two uniform distribution, what about 3 or ...
0
votes
1answer
25 views

Suppose that $E$ and $F$ are two events? [closed]

Suppose that $E$ and $F$ are two events and that $P(E\cap F)= 0.4$ and $P(E)= 0.8$. What is $P(F\mid E)$ ?
2
votes
2answers
29 views

Probability Distribution

I'm thinking about a set of n users on Facebook. Between each of the $\binom{n}{2}$ pairs of distinct friends, lets say an edge (indicating that the two people are friends) is independently present ...
2
votes
0answers
84 views

Of strings and substrings: A problem of probability

Problem Let $\Sigma=\{a, b\}$. Let $\Sigma^*$ denote the Kleene star of $\Sigma$: \begin{equation*} \Sigma^* = \{\varepsilon, a, b, aa, ab, ba, bb, aaa, aab, \ldots\} \end{equation*} where ...
1
vote
1answer
17 views

Reconstruct multivariate binary distribution from marginals

I'm have a random vector $\bf a$ with binary entries, $a_i \in \{0,1\}$. The probability distribution $P({\bf a})$ is not fully specified, but I have the marginals $p_i$, which are the probabilities ...
0
votes
2answers
25 views

how to estimate parameters of a triangular distribution? [closed]

I have a set of observations, and they come from a triangular distribution. Now I want to estimate its parameters, but how?
0
votes
1answer
17 views

Geometric distribution with given probability value.

The probability of a man hitting a target is $2/3$. If he doesn't stop shooting until he hits the target for the first time, a) What is the probability of taking 5 shots to hit the target? b) Which is ...
1
vote
1answer
19 views

What is the “Cumulative Distribution of the magnitude of the N-dimensional standard gaussian”

I am confused by this line from a paper: "Let $F_1(x)$ be the cumulative distribution of the magnitude of an $n$−dimensional standard Gaussian random variable and $F_2(x)$ be the cumulative ...
2
votes
1answer
20 views

Properties of unimodal functions

A probability density function $f$ is said to be unimodal if the value at which the maximum value of the function is attained is unique. I am reading some papers on econometrics that appear to use ...
1
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2answers
36 views

Question about probability distributions

I've recently came across this question: ...
0
votes
1answer
38 views

Infrequent fail of the popular parameter estimators, having several beta-distributed random variables to be estimated

I have a project in which there exist $N$ Beta-distributed Random variables each of which should be estimated, having a sample for each of them. The sample domain is $\{0.1,0.3,0.5,0.7,0.9\}$ and the ...
0
votes
0answers
33 views

How do I calculate conditional PDF?

Obtain $$P(2 < Y < 3 | X = 1)$$ where the joint pdf of X and Y is $$f_{X,Y}(x,y) = (6-x-y)/8$$ where $$0 < x < 2$$ and $$2 < y < 4$$? so first, I did $$f_Y|X=1(y) = ...