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

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2answers
27 views

For a non-negative absolutely continuous random variable $X$, with distribution $F$. Why is $\lim_{t\rightarrow \infty}t(1-F(t))=0$?

So I am given a non-negative absolutely continuous random variable $X$ with distribution $F$, and density $p_X$. I am given the definition of expectation using simple functions and the survival ...
1
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1answer
1k views

Understanding the difference between normal distribution and lognormal distribution

I'm having trouble understanding the difference between a normal distribution and lognormal distribution. Here's what I've done so far. Definitions of lognormal curves: "A continuous distribution in ...
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0answers
21 views

CDF of ratio of Gamma distribution with different parameters

Let $X$ be gamma distributed random variable with parameters $a$ and $b$. Let $W$ be gamma distributed random variable with parameters $c$ and $d$, such that \begin{equation} f_X(x) = ...
-1
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0answers
13 views

statistics uniform distribution [on hold]

Dominic released his rabbit to roam on the lawn; some time later, it returned and so he continued doing that daily. Over time, he found a pattern of the time T (in hours) Rabbit stayed outside. If ...
0
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1answer
10 views

Joint CDF from conditional cdf

I would like to derive an expression of the following joint CDF $P[X \leq x,Y \leq y]$ based on the conditional CDF $P[X \leq x | Y=y]$ and the pdf $P[Y=y]$ that are considered to be known. I get a ...
0
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1answer
14 views

A question of joint CDF

I am confused about how to use the joint probabilities to find the joint CDF.
0
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0answers
9 views

Probability that one random variable is greater than or equal to another

Assume $X$ and $Y$ are i.i.d. with exponential distribution with parameter $\lambda = 1$ (the probability density functions $p_X (x) = e^{-x}$ and $p_Y (y) = e^{-x}$ in $[0, +\infty)$, $0$ otherwise). ...
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0answers
2 views

Unable to follow notation and meaning of probability distribution for binary time series

I am unable to understand concepts related to the probability distribution of binary time series. [Mathematics is not my background]. This is from the book Binary time series by Benjamin Kedem, vol ...
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1answer
13 views

Probability of Normal Distribution

Let's say that 10 sumo wrestlers were to squeeze into an elevator that could only hold a max capacity of 2300 pounds. Let's say that the weight of the sumo wrestlers is normally distributed with a ...
1
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1answer
15 views

A question of Joint PDF

I have not idea about part a. I know I need to prove the integration of f(x,y)=1, but how should I deal with the range of x and y.
0
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1answer
30 views

Is $r_2$ a uniformly at random value in $Z_n$, where $r_2=r_1 . m$

Let $m$ be an arbitrary value in $Z_n$, where n is RSA modulo (n=p.q, where p and q are large primes). Then have: $r_2=r_1 . m$, where $r_1$ is a value chosen uniformly at random : $r_1\in Z^*_n$. ...
1
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1answer
17 views

Elementary Probability: Expected Value

I must say, first, that this question IS a homework assignment and I do not wish an answer here, for I already posssess it. I want to know if there is a general procedure of simplification in this ...
2
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1answer
137 views

Coupling Pairs of Random Variable.

Let $\{X_i\}_{i=1}^{n}$ and $\{Z_i\}_{i=1}^{n}$ be sets of independent random variables with coupling $\{X^{\hat{}}_i\}_{i=1}^{n}$, $\{Z^{\hat{}}_i\}_{i=1}^{n}$ respectively. It then states ...
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0answers
7 views

Deriving a simple PDF

I am looking for deriving the pdf of $Z$ where $Z= (\sum\limits_{i=1}^N a_i X_i +Y_1)^2 + (\sum\limits_{i=1}^N b_i X_i +Y_2)^2$, where $X_i$ and $Y_i$ are independent, zero mean Gaussian random ...
-1
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1answer
25 views

Cumulative distribution function of exponentials

I have the cumulative distribution function $F(x)=(1-e^{-x})\mathbb{1}_{x≥0}$ and want to write the CDF to $F(\frac{x-\mu}{\sigma})$. I have derived ...
1
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1answer
323 views

Solving Probability Density Function for continuous random variable

The probability density of a random variable $x$ is $$f(x)=a\ \cdotp x^2\ \cdotp \mathrm{e}^{−kx}\ (k>0,\ 0\leq x\leq \infty)$$ Then, the coefficient $a$ equals $$(i)\frac{k^3}{2}\ \ \ \ (ii)\ k^3 ...
1
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0answers
36 views

How to demonstrate a particular functional equation solution

In order to find a prior probability distribution I have to solve the following functional equation: $$af\left(\frac{a\theta}{1-\theta-a\theta}\right)=(1-\theta+a\theta)^2f(\theta)$$ the solution of ...
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1answer
16 views

Conditional Expectation of Binomial Given $X \leq x$

Are there any neat formulas to reduce something like $\sum_{i=0}^{x} i \binom{n}{i} p^i (1-p)^{n-i}$ where $x<n$? This would be proportional to $\mathbb{E}(X\leq x)$ where $X$~$\text{Bin}(n,p)$. ...
0
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1answer
66 views

Probability in DNA segmentation

I have formulated these questions ss part of a research in medical science (DNA segmentation): A series of $M$ identical balls is arranged on a line. A partition is formed by placing a stick to ...
1
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2answers
54 views

Variance deduction

the definition of variance is $V(X) = E((X-E(X))^2 )$ For a discrete random variable: if we have put $Y = g(X)$ , where $g$ is a real function $E(Y) = E(g(X)) = \sum\limits_{k} g(k)p_X(k)$ , ...
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0answers
21 views

The Joint PDF question [on hold]

Can someone help me do this series of question? I really need help and I have no idea about it. I have no idea of how to deal with the range of x and y.
1
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2answers
16 views

Probability Distribution Function?

An urn contains 8 green balls and 17 yellow balls. A ball is drawn from the urn and its color is noted and then the ball is placed back in the urn. 5 balls are drawn this way. Let $X$ denote the ...
1
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1answer
29 views

Prove that $ \lim_{n \to \infty} \frac{\Phi(- \sqrt{n})}{f(\sqrt{n})} = 1$.

Let $X$ be standard normal random variable. Let $\Phi$ be a distribution function of $X$ and $f$ density function of $X$. Prove that $$ \lim_{n \to \infty} \frac{\Phi(- \sqrt{n})}{f(\sqrt{n})} = 1$$. ...
0
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1answer
24 views

Negative binomial with conditional probability

Let X be a random variable that follows a negative binomial distribution: NB(r=4, p=0.4) Calculate P(X = 8 | x > 6) I know how to calculate P(X = 8): $$ \binom{7}{3} \cdot (1 - 0.4)^{7-3} \cdot ...
2
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1answer
66 views
+100

A Law of Large Numbers Without Replacement

Let $(n_1,...,n_r)$ be $r$ positive integers, and let $n=n_1+...+n_r$. Fo each positive integer $m$ consider an urn containing $mn$ balls, of which $mn_1$ are of type 1,..., $mn_r$ of type r. For each ...
1
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1answer
19 views

How to set up problem involving Poisson RV

Consider an example where customers entering a store is a Poisson random variable with $\lambda=15$. How do you find the probability that 100 or fewer people will walk into the store in any five-day ...
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0answers
21 views

Easy question: Multiple random variables vs. product of probability spaces

I never had a course in probability theory and the definitions we work with are quite informal, so I am a little bit confused about the difference between "multiple" random variables and the notion of ...
0
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1answer
21 views

Using the binomial distribution as the distribution for a sum of Bernoulli random variables?

Knowing that the sum of $n$ independent Bernoulli random variables with parameter $p$ ($p \in (0,1)$) has a binomial distribution $Bin(n,p)$, how can I use the Central Limit Theorem (or any other ...
0
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0answers
15 views

How to find the distribution of $W = (X_1^2 + X_2^2) / 4$, where both $X_i$ are iid with given moment generating function?

$X_i$s are i.i.d random variables. (The number of random variables is not specified.) And m.g.f of X is $M(t) = \exp[2(t^2)]$. How can I get the distribution of new random variable $W = (X_1^2 ...
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1answer
24 views

Finding the distribution of $Y_2$,knowing that $Y_1 \in Po(\lambda/2)$

The random variables $N,X_1,X_2..$ are independent, $N\in Po(\lambda)$, and $X_k \in B(1/2) , k \geq 1$ Set. $Y_1 =\sum\limits_{k=1}^{N}X_k $ and $Y_2 = N - Y_1$. Determine the distributions of ...
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0answers
22 views

Lottery probability using hypergeometric distribution

Let's say that we're interested in a "powerball"-type lottery system where five balls are drawn at random and without replacement from an urn containing 25 labeled balls. A players pays $1 to guess ...
0
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0answers
19 views

joint distribution of two arbitrary distributions?

F = S + E where S: start time and E: execution time, which are arbitrary probability distributions. S and E are discrete and independent.F is finish time of a task which starts in random start time ...
1
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1answer
98 views

What to do with a random variable when we know its mean and variance but does not know which distribution it is?

Let Y be a random variable with mean μ and variance σ^2 where the support is (0, ∞). Suppose you are offered to play a game where you choose a number z between (0, ∞). If a realization of the random ...
0
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1answer
10 views

Limiting random variable

My question is about limiting r.v. Suppose, we have a sequence of r.v.s. $\{X_n\}$. And we know that $\liminf X_n=-\infty$ and $\limsup X_n =\infty$ almost surely. What can we say about $\lim X_n$. ...
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1answer
25 views

Question in permutations

When we use this law? And in any case we use it? Thank you and I wish clarification.
1
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1answer
35 views

Applying the Poisson Distribution to problems

The number of traffic accidents at a certain intersection is thought to be well modeled by a Poisson process with a mean of $3$ accidents per year. Find the probability that more than one year ...
3
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1answer
16 views

Distribution of numbers $n = \max(x, y)$ where $x, y$ are random numbers between $0$ and $1$

I define a function $$f = \max(\mbox{rand}(0, 1), \mbox{rand}(0, 1))$$ such that $f$ returns the maximum (greater number) of two random selected numbers between $0$ and $1$. Plotting a histogram for ...
2
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0answers
13 views

How to compute uniformly distributed points on an ellipse

The ellipse can be parametrized in polar coordinates by $$r(\theta)=\frac{1}{a+\cos\theta}$$ up to a scaling factor, and $a>1$. Suppose we measure $S$, the distance along the ellipse from the ...
1
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0answers
9 views

Sampling from copula

I'm trying to understand the general technique for sampling from a copula. I understand that, if $X$ and $Y$ have known marginals, their copula is defined to be, $$ ...
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0answers
17 views

Density function of $Y $given that $ Y = 2X, X \leq 2, Y = X^2, X > 2.$

So we have $X$ with density $$f_X(x) = 1/x^2, x \geq 1, f_X(x) = 0, x < 1.$$ And $$Y = 2X, X \leq 2, Y = X^2, x > 2.$$ So I drew my graphs of $f_X$ and $Y$, but where the functions change is ...
5
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2answers
71 views

“Back to square one” problem

There's a problem I've been stuck on in preparation for junior programming contest I'm going to participate in. It is as follows: The "back to square one" problem is played on a board that has ...
0
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0answers
20 views

Distribute Chocolates among m children Non Uniformly

Suppose person B has N chocolates,He has to distribute these chocolates non-uniformly among M children. Suppose child mi has weight Wi and the total weight is Wt. One way to do so is to distribute ...
0
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0answers
9 views

Negative binomial distribution's mgf!

I've got problem. during my trying to find mgf(moment generating function) of negative binomial distribution, I found myself cannot understanding process. How does second line become third line? ...
2
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1answer
67 views

Confidence interval for the conversion on site

I am the developer of web service and I'm trying to to build some plots for the inner dashboard. I raised two questions that I can not solve on their own. Suppose ...
0
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1answer
34 views

Battery lifetime as normal distribution?

I want to model battery lifetime, which decrements continuously at every epoch (i.e., work-cycle) in the following way. So it takes values such as 100, 99.7, 99.3, 99.2, ... 0 (a continuous random ...
1
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1answer
14 views

Find the conditional distribution ,which probably is binomial

Sheila has a coin with $P(head)= p_1$ and betty has a coin with $P(head) = p_2$.Sheila tosses her coin $m$ times. Each time she obtains heads , betty tosses her coin(otherwise not). Find the ...
0
votes
1answer
24 views

Binomial Probability (Dice)

The throwing of a one or two is called a success. The six dice are thrown together 64 times and the frequencies of the throws with 0, 1, 2,..., 6 successes are summed over all six pairs are as ...
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0answers
21 views

Histogram Interpretation

For some reason my brain cannot comprehend this histogram. Is this normal or skewed? The main portion of the histogram looks like a normal distribution, but it is not placed in the center, and there ...
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1answer
13 views

probability distribution

I would really be grateful if someone could answer me promptly. I believe i should use the poisson distribution model because that is the suitable one however i cannot satisfy the condition of ...
0
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2answers
27 views

Expected value for number

Suppose you have a game with $n$ stages. For every stage $i$, you have $p(i)$ probability to advance to the next stage, and $1-p(i)$ probability to return to stage $1$. You win the game by advancing ...