Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

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.

1
vote
1answer
328 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 ...
0
votes
1answer
148 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
152 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
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
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
1answer
251 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
2answers
169 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(...
28
votes
2answers
23k 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 ...
28
votes
2answers
29k 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
5k 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 $...
20
votes
3answers
34k 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 ...
12
votes
1answer
11k views

Sum of Bernoulli random variables with different success probabilities

Let $X_{i} \in \{0,1\}$ be Bernouli random variable with probability of success $p_{i}$, i.e., $P(X_{i}=1) = p_{i}$ and $P(X_{i}=0) = 1-p_{i}$ and let $Y=\sum_{i=1}^{n}X_{i}$ for $n>0$. Is it ...
19
votes
4answers
69k 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
2k 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\...
8
votes
4answers
12k 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 ...
13
votes
2answers
17k 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? After having scanned ...
11
votes
1answer
3k 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
13k 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{...
9
votes
4answers
202 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)?
6
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, \...
6
votes
2answers
16k 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} \...
4
votes
1answer
453 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
1k 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{...
4
votes
1answer
4k views

Uniformly Most Powerful Test for a Uniform Sample

Let $X_{1}, \dots, X_{n}$ be a sample from $U(0,\theta), \theta > 0$ (uniform distribution). Show that the test: $\phi_{1}(x_{1},\dots,x_{n})=\begin{cases} 1 &\mbox{if } \max(x_{1},\dots,x_{n})...
2
votes
2answers
12k views

Determine the PDF of $Z = XY$ when the joint pdf of $X$ and $Y$ is given

The joint probability density function of random variables $ X$ and $ Y$ is given by $$p_{XY}(x,y)= \begin{cases} & 2(1-x)\,\,\,\,\,\,\text{if}\,\,\,0<x \le 1, 0 \le y \le 1 \\ & \,0\,\,\,\...
9
votes
4answers
19k views

Fisher information of a Binomial distribution

The Fisher information is defined as $\mathbb{E}\Bigg( \frac{d \log f(p,x)}{dp} \Bigg)^2$, where $f(p,x)={{n}\choose{x}} p^x (1-p)^{n-x}$ for a Binomial distribution. The derivative of the log-...
6
votes
5answers
1k views

Product of cdf and pdf of normal distribution.

A continuos random variable $X$ has the density $$ f(x) = 2\phi(x)\Phi(x), ~x\in\mathbb{R} $$ then (A) $E(X) > 0$ (B) $E(X) < 0$ (C) $P(X\leq 0) > 0.5$ (D) $P(X\ge0) &...
5
votes
3answers
630 views

How to calculate $E[(\int_0^t{W_sds})^n], n \geq 2$

Let $W_t$ be a standard one dimension Brownian Motion with $W_0=0$ and $X_t=\int_0^t{W_sds}$. With the help of ito formula, we could get $$E[(X_t)^2]=\frac{1}{3}t^3$$ $$E[(X_t)^3]=0$$ When I try to ...
5
votes
1answer
2k views

Distribution of weighted sum of Bernoulli RVs

Let $x_1,...,x_m$ be drawn from independent Bernoulli distributions with parameters $p_1,...,p_m$. I'm interested in distribution of $t=\sum_i a_ix_i,~a_i\in \mathbb{R}$ $m$ is not large so I can ...
5
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 $...
3
votes
2answers
7k views

First exit time for Brownian motion without drift

I am dealing with the simulation of particles exhibiting Brownian motion without drift, currently by updating the position in given time steps $\Delta t$ by random displacement in each direction drawn ...
1
vote
1answer
582 views

Compound Distribution — Normal Distribution with Log Normally Distributed Variance

Could someone please point me to a source or suggest ways in which we can obtain the Distribution, Density Functions, Expected Value, etc. of a Normal Distribution whose variance is distributed Log ...
12
votes
0answers
8k views

Uniform distribution on the surface of unit sphere

It is known that given $X=(X_1, X_2, \ldots, X_n)$ iid $\sim N(0,1)$, then $X/\sqrt{X_1^2+\cdots+X_n^2}$ is uniformly distributed on the surface of unit sphere. Intuitively, I know that that's ...
7
votes
1answer
3k views

Geometric distribution with unequal probabilities for trials

I am researching an engineering problem in which I want to model the probability distribution of the number X of independent trials needed to get one success. If the probability of success at each ...
6
votes
2answers
3k views

Showing $\cos(t^2)$ is not a Characteristic Function

Usually when we try to show a function is not a characteristic function, we would prove it is not uniformly continuous. I am wondering if there is any other way to show $\cos(t^2)$ is not a ...
4
votes
1answer
2k views

Convolution of two Uniform random variables

We have $X \sim \mathrm{Unif}[0,2]$ and $Y \sim \mathrm{Unif}[3,4]$. The random variables $X,Y$ are independent. We define a random variable $Z = X + Y$ and want to find the PDF of $Z$ using ...
3
votes
3answers
1k views

Distribution of the sum of $N$ loaded dice rolls

I would like to calculate the probability distribution of the sum of all the faces of $N$ dice rolls. The face probabilities ${p_i}$ are know, but are not $1 \over 6$. I have found answers for the ...
3
votes
1answer
798 views

$E|X-m|$ is minimised at $m$=median

For a continuous random variable $X$, I want to show that $E|X-m|$ is minimum implies $m$ is the median of the distribution. Assume that the distribution function is $F$ and the density function is $f$...
2
votes
2answers
313 views

$E_n =\lbrace X_n > X_m \ \forall m < n \rbrace $ are independent

I'm stuck with this exercise. Suppose $(X_n)$ are independent random variables defined on $(\Omega, \mathfrak{F}, P)$ with the same p.d.f. Let $E_1 = \Omega$ and for $n \geq 2$ $$E_n =\lbrace X_n &...
9
votes
2answers
9k views

What is the PDF of random variable Z=XY?

Given two independent random variables X and Y, how can I find the PDF of random variable $Z=XY$? *If their joint distribution is required, assume that we also have it.
8
votes
1answer
477 views

Estimates for the normal approximation of the binomial distribution

I'm interested in estimates of the normal approximation for binomial distributions, i.e. in estimates of $$\sup_{x\in\mathbb R}\left|P\left(\frac{B(p,n)-np}{\sqrt{npq}} \le x\right) - \Phi(x)\right|$$...
8
votes
2answers
1k views

Name/significance of integral of the square of a probability density function

Background/Motivation Given a probability density function $f(x)$, the mean of the corresponding random variable is the $x$-coordinate of the centroid of the region under the graph of $f$. I wondered:...
8
votes
1answer
15k views

Expected value of normal distribution given that distribution is positive

Given $X \sim N(0, \sigma^2)$ (that is, $X:\mathbb{R} \to \mathbb{R}$ is a normal random variable with mean $0$ and variance $\sigma^2$), I'm trying to calculate the expected value of $X$ given that $...
8
votes
3answers
5k views

Expected value as integral of survival function

Let $T$ be a positive random variable, $S(t)=P(T\geq t)$. Prove that $$E[T]=\int^\infty_0 S(t)dt.$$ I have tried this unsuccessfully.
8
votes
2answers
7k views

Poisson distribution with exponential parameter

I don't know how to solve Exercise 8, Section 5.2 from Geoffrey G. Grimmett, David R. Stirzaker, Probability and Random Processes, Oxford University Press 2001. For those who don't have this book: ...
7
votes
2answers
701 views

A reference for a Gaussian inequality ($\mathbb{E} \max_i X_i$)

I am looking for a reference to cite, for the following "folklore" asymptotic behaviour of the maximum of $n$ independent Gaussian real-valued random variables $X_1,\dots, X_n\sim \mathcal{N}(0,\sigma)...
7
votes
1answer
659 views

Distribution of a random real with i.i.d. Bernoulli(p) binary digits?

Let $X_1, X_2, X_3, \ldots$ be an infinite sequence of i.i.d. Bernoulli($p$) random variables, and define the random real number $X = (0.X_1X_2X_3\ldots)_2$. Question(s): What can be proved about ...
6
votes
2answers
119 views

Distribution at First Time a Sum Reaches a Threshold

Consider the following problem. Roll a die many times, and stop when the total exceeds $M$, for some prescribed threshold $M$. Call this time $\tau$, and call the running score after $n$ rolls $X_n$. ...
6
votes
2answers
13k views

Sum of n i.i.d Beta-distributed variables

Assume that I have $n$ variables that are each $X_i \sim \text{Beta}(\alpha, 1)$ distributed (with the same $\alpha$, i.i.d.). Is there anything known about the distribution of the sum $Y=\sum_i X_i$?...