Questions tagged [probability-distributions]

Questions on using, finding, or otherwise relating to probability distributions, probability density functions (pdfs), cumulative distribution functions (cdfs), or other related functions.

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6
votes
1answer
135 views

Help with convergence in distribution

$Y$ is a random variable with $$M(t) = \frac{1}{(2-\exp(t))^s}.$$ Does $$\frac{Y-E(Y)}{\sqrt{\operatorname{Var}(Y)}}$$ converge in distribution as $s$ tends to infinity? I let $Z = \frac{Y-E(Y)}{\...
3
votes
1answer
899 views

Proof that sum of independent normals is normal using convolutions

Let $X, Y$ be independent standard normal random variables. We already know that $X+Y$ is normal with mean 0 and variance 2. However, I am trying to prove this result in a slightly different way than ...
3
votes
2answers
6k views

Finding the mean distance between n points evenly distributed in a disc of radius r

In reading this article about updated estimates for the number of exoplanets in the Milky Way, I am curious how to get an estimate of the mean distance between them. The Milky Way is ~50,000 light ...
3
votes
1answer
3k views

MLE (Maximum Likelihood Estimator) of Beta Distribution

Let $X_1,\ldots,X_n$ be i.i.d. random variables with a common density function given by: $f(x\mid\theta)=\theta x^{\theta-1}$ for $x\in[0,1]$ and $\theta>0$. Clearly this is a $\operatorname{...
3
votes
2answers
300 views

A basic doubt on joint distribution

How to calculate the following probability $P(X \leq x, Y=y)$ where $X$ is a continuous random variable and $Y$ is a discrete random variable. I have been given the distribution of $X$ and ...
3
votes
2answers
85 views

Is there any way to calculate harmonic or geometric mean having probability density function?

I have probability density of function of some data (it's triangular.) How can I calculate harmonic or geometric mean of the data? I know for calculating arithmetic mean of a variable like $K$, I have ...
3
votes
2answers
260 views

Prove independence of events given random variables are iid and have continuous cdf

Let $Y_1, Y_2, \ldots$ be independent and identically distributed random variables in $(\Omega, \mathscr{F}, \mathbb{P})$ s.t. their distributions are continuous and $$F_Y(y) := F_{Y_1}(y) = F_{Y_2}(y)...
3
votes
0answers
156 views

Approximate a density function from sampled data

Let $(E,\mathcal E,\mu)$ be a measure space $E_0\in\mathcal E$ with $\mu(E_0)\in(0,\infty)$ and $\mathcal E_0\subseteq\left.\mathcal E\right|_{E_0}$ be finite and disjoint with $$E_0=\biguplus\...
2
votes
1answer
1k views

Probability of throwing balls into bins

You are throwing n balls into m bins randomly. What is the probability to be empty of the first $k$ bin? Given $k$ bins are empty. What is the probability to be empty of $(k+1)th$ bin? Forget the ...
2
votes
1answer
2k views

distribution of the normal cdf

I am wondering what is the probability density function for the normal cdf $\Phi (aX+b)$, where $\phi$ is the usual standard normal cumulative distribution function I want to calculate $\mathbb{E}[\...
2
votes
3answers
393 views

Probability: Permutations

Consider the experiment of picking a random permutation $\pi$ on $\{1,2,...,n\}$, and define the associated random variable $f(\pi)$ as the number of fixed points of $\pi$, i.e, the number of $i$ such ...
1
vote
1answer
1k views

probability question on characteristic function

I got a big problem with my exam practice question on characteristic function. Here are two. Let $X$, $Y$ be two independent random variables with the following characteristic functions: $$\...
1
vote
1answer
55 views

Is it true that $\lim\limits_{x\to\infty}{x·P[X>x]}=0$?

I ask this because I'm trying to understand a proof of the expected value of a non negative random variable is equal to $\int_{0}^{\infty}(1-F(x))dx$ ($F$ is the distribution function of $X$) when $E[...
1
vote
2answers
2k views

Show that $S = \sqrt{S^2}$ is a biased estimator of $\sigma$ given a random sample from a normal distribution …

Suppose $Y_1, \ldots, Y_n$ is a random sample from a normal distribution with mean $\mu$ and variance $\sigma^2$. Let $S^2$ be the sample variance, which is unbiased for $\sigma^2$. GOAL: Show that $...
1
vote
1answer
112 views

Find the distribution of sum and product of standard normal random variables

Let $X,Y$ and $Z$ be three independent real valued random variables. All with finite second moment and all with mean $0$ and variance $1$. Define $$ W= \frac{X+YZ}{\sqrt{1+Z^2}} $$ Find the ...
1
vote
1answer
4k views

What is the distribution of $Y = e^X$ when $X$ is normal?

What is the distribution of $Y = e^X$ when $X$ is normally distributed? Am I supposed to use characteristics function of normal random variable ?
1
vote
1answer
349 views

Probability of success on third and fourth trials?

I have a pretty basic probability question, but I'm just having difficulties remembering what distribution this is. A coin is tossed until a head appears two times in a row. Given that we are using a ...
1
vote
3answers
511 views

Zero integral of measurable $f$ on every interval implies $f=0$?

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a Borel measurable function, such that $\displaystyle \int_{a}^{b}f(x)dx=0$ for every $a<b$ in $\mathbb{R}.$ Is it true that $f(x)=0$ for every $x\in \...
0
votes
1answer
35 views

Is $v=\ (r_i)^{-1}\cdot z $, a uniformly random value of a field?

We consider a finite field $\mathbb{F}_q$ where $q=2p+1$ and $q$ and $p$ are prime numbers. Let $r_i$ be a value picked uniformly at random from the field such that $r_i>\frac{q}{2}$. Let $z$ be a ...
0
votes
1answer
153 views

Average number of terms required in a sum of exponential variables to reach a specific limit

I have a sum $Y=\sum_{i=1}^{\infty}(X_i-t)u(X_i-t)$ where all $X_i's$ are i.i.d exponentially distributed random variables with parameter $\lambda$ and $t$ is a constant. I want to know how many term ...
0
votes
2answers
2k views

Conditional distribution of geometric variables

Setting Suppose X1 and X2 are independent with the common geometric distribution w(k; p). Determine the conditional distribution of X1 given that X1 + X2 = n. Solution My argument is $$\Pr[X_1| ...
0
votes
0answers
48 views

Specify the area of interest to find the marginal pdf

Suppose the following joint pdf: $f_{X,Y}(x,y) = \frac{3}{2}x$ if $1\leq x \leq 2$ and $0 \leq y \leq x$. Note that $f_{X,Y}(x,y) = 0$ otherwise. I am told to calculate the marginal pdf for $Y$, ...
0
votes
1answer
4k views

Joint cdf and pdf of the max and min of independent exponential RVs [duplicate]

Let $X$ and $Y$ be independent random variables. Each has an exponential distribution with parameter $\lambda$. Define two new random variables by $W = \min({X,Y}) $ $Z = \max({X,Y})$ Find the ...
0
votes
2answers
724 views

Joint density for exponential distribution

Let $X_1$ and $X_2$ be independent random variables each having a exponential distribution with mean $\lambda = 1$. (a) Find the joint density of $Y_1 = X_1$ and $Y_2 = X_1 + X_2$. (b) Get the ...
0
votes
1answer
144 views

Strong law of large numbers for a scaled sequence of normally distributed random variables

Let $f\in C^3(\mathbb R)$ be positive $g:=\ln f$ $d\in\mathbb N$, $$p_d(x):=\prod_{i=1}^df(x_i)\;\;\;\text{for }x\in\mathbb R^d$$ and $\lambda^d$ denote the Lebesgue measure on $\mathcal B(\mathbb R^...
0
votes
2answers
174 views

Find the cumulative function and the density of $m_n$ and $M_n$, where $m_n=\min(X_1,X_2,…,X_n)$ and $M_n=\max(X_1,X_2,…,X_n)$. [closed]

Let be $X_1,X_2,...,X_n$ be uniformly distributed random variables i.i.d. a) Find the cumulative function and the density of $m_n \text{ and } M_n$ , where $m_n=min(X_1,X_2,...,X_n)$ and $M_n=max(...
0
votes
2answers
165 views

How to show a binomial random variable dominates another binomial random variable with a smaller success value?

Let $X\sim B(n,p_h)$ and $Y\sim B(n,p_\ell)$ be two random variables following a respective binomial distribution, where $p_h>p_\ell$. I want to show that $$P(X\ge\alpha)\ge P(Y\ge\alpha),$$ for ...
0
votes
0answers
36 views

Are there lots of various Empirical distribution

Chapter 3 of the deeplearningbook gives this formula for "Empirical distribution" \begin{equation} \hat{p}(x) = \frac{1}{m} \sum_{i=1}^m \delta(x - x^{(i)}) \tag{3.28} \end{equation} Wiki gives this ...
0
votes
1answer
348 views

Mixed Probability Distribution - Expected Value

Consider for one car owner the insurance policy with the following clauses: Deductible: If the loss $X>d$, then the insurer pays only for loss above $d>0$. Coverage Limit: If the loss $X>l$, ...
0
votes
1answer
172 views

Find the correlation coefficient $\rho_g$ of $(G(x), H(y))$

Find the correlation coefficient $\rho_g$ of $(G(x), H(y))$ where $G(x),H(y)$ are marginal C.D.Fs of $X,Y$ and $X,Y$ follow bivariate normal distribution $\text{B.N.}(\mu_1,\mu_2,\sigma_1,\sigma_2,\...
29
votes
2answers
25k views

How is logistic loss and cross-entropy related?

I found that Kullback-Leibler loss, log-loss or cross-entropy is the same loss function. Is the logistic-loss function used in logistic regression equivalent to the cross-entropy function? If yes, can ...
24
votes
3answers
7k views

Intuition for probability density function as a Radon-Nikodym derivative

If someone asked me what it meant for $X$ to be standard normally distributed, I would tell them it means $X$ has probability density function $f(x) = \frac{1}{\sqrt{2\pi}}\mathrm e^{-x^2/2}$ for all $...
28
votes
2answers
31k views

Expectation of the min of two independent random variables?

How do you compute the minimum of two independent random variables in the general case ? In the particular case there would be two uniform variables with a difference support, how should one proceed ?...
21
votes
3answers
38k views

Maximum Likelihood Estimator of parameters of multinomial distribution

Suppose that 50 measuring scales made by a machine are selected at random from the production of the machine and their lengths and widths are measured. It was found that 45 had both measurements ...
20
votes
4answers
77k views

Multiplication of a random variable with constant

Suppose $X$ is a random variable which follows standard normal distribution then how is $KX$ ($K$ is constant) defined. Why does it follow a normal distribution with mean $0$ and variance $K^2$. ...
10
votes
2answers
3k views

Prove that the maximum of $n$ independent standard normal random variables, is asymptotically equivalent to $\sqrt{2\log n}$ almost surely.

Lets $(X_n)_{n\in\mathbb{N}}$ be an iid sequence of standard normal random variables. Define $$M_n=\max_{1\leq i\leq n} X_i.$$ Prove that $$\lim_{n\rightarrow\infty} \frac{M_n}{\sqrt{2\log n}}=1\quad\...
9
votes
4answers
13k views

Showing that ${\rm E}[X]=\sum_{k=0}^\infty P(X>k)$ for a discrete random variable

Let $X$ be a discrete random variable whose range is $0,1,2,3,\ldots$. Prove that $$ {\rm E}[X]=\sum_{k=0}^\infty P(X>k). $$ How to prove this? I tried a bit but unable to post due to formatting ...
7
votes
2answers
4k views

Proof of the affine property of normal distribution for a landscape matrix

The widely used/mentioned/assumed affine property of multivariate normal distributions says that: Given a random vector $x \in R^N$ with a multivariate normal distribution -- $x \sim N_x(\mu_x, \...
14
votes
3answers
19k views

PDF of product of variables

Could anyone please indicate a general strategy (if there is any) to get the PDF (or CDF) of the product of two random variables, each having known distributions and limits? My particular need is ...
11
votes
1answer
4k views

Why left continuity does not hold in general for cumulative distribution functions?

Definition: The c.d.f. $F$ of a random variable $X$ is a function defined for each real number $x$ as follows:$$F(x)=\Pr(X\leq x) \text{ for } -\infty<x<\infty$$ Let $$F(x^-)=\lim_{y\rightarrow ...
4
votes
1answer
15k views

Probability density function of $max(X,Y)$

Assume that we have a random variable $W = \max({X,Y})$ and that we would like to find the pdf of $W$. This is what I have done. $$ F_W(w)= \mathbb{P}[ W\leq w]=\mathbb{P}[ \max({X,Y})\leq w]=\mathbb{...
13
votes
2answers
7k views

joint distribution, discrete and continuous random variables

This may be trivial, but if X is a random variable uniformly distributed over $[0,1]$ and Y is a discrete random variable such that $\mathbb{P} (Y=y_1) = \lambda \in (0,1]$ and $\mathbb{P} (Y=y_2) = 1 ...
9
votes
4answers
241 views

An urn has 4 balls of 4 different colours Red,Blue,Green,Yellow.

An urn has $4$ balls of $4$ different colours; red, blue, green, and yellow. I pick one ball at random at first and if it is red, I paint it blue and return it to the urn. If it is blue, I paint ...
8
votes
1answer
5k views

Upper/lower bound on covariance two dependent random random variables.

X and Y are two dependent random variables. Marginal pmfs f(X) and f(Y) is given, but joint pmf f(X,Y) is not known. Is it possible to find upper/lower bound on covariance cov(X,Y)?
7
votes
2answers
17k views

What is the difference between a Poisson and an Exponential distribution?

For a Poisson distribution: $$\mathsf{P}(X=x)=\frac{e^{-\mu}\times \mu^x}{x!}$$ where $\mu$ is the mean number of occurrences. For an Exponential distribution: $$f(x;\lambda) = \begin{cases} \...
7
votes
3answers
2k views

What is the intuition behind the exponential distribution?

My textbook gives the definition of the exponential distribution: $$f(x) = \lambda e^{- \lambda x}$$ But I can't find a good explanation online about how this was derived/where it comes from, or the ...
6
votes
2answers
4k views

compound of gamma and exponential distribution

What is the distribution of a exponential distribution, whose parameter is drawn form the gamma distribution $$ X \sim \operatorname{Gamma}(\alpha,\beta)$$ $$ Y \sim \operatorname{Exp}(X)$$ how is $...
5
votes
2answers
7k views

Distribution of $\max(X_i)\mid\min(X_i)$ when $X_i$ are i.i.d uniform random variables

If I have $n$ independent, identically distributed uniform $(a,b)$ random variables, why is this true: $$ \max(x_i) \mid \min(x_i) \sim \mathrm{Uniform}(\min(x_i),b) $$ I agree that the probability ...
4
votes
1answer
477 views

Conditional return time of simple random walk

Consider a simple symmetric random walk on $\mathbb{Z}$, $(S_t)_{t \geq 0}$. Call $\tau_k = \min\{t \in \mathbb{N}\, : \, \, S_t =k \}$, the hitting time of $k \in \mathbb{N}$. Call $\tau^* = \min\{t &...
4
votes
3answers
2k views

Random variables defined on the same probability space with different distributions

Consider the real-valued random variable $X$ and suppose it is defined on the probability space $(\Omega, \mathcal{A}, \mathbb{P})$. Assume that $X \sim N(\mu, \sigma^2)$. This means that $$ (1)\text{...