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

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151 views

Statistics Poisson Distribution

Suppose that the number of eggs laid by a certain insect has a Poisson distribution with mean λ. The probability that any one of the eggs hatches is p. Assume that the eggs hatch independently of one ...
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1answer
190 views

Sampling without replacement until object found

I'm wondering about sampling without replacement until an object is found. I can't seem to wrap my head around it. The random variable I want to use is X which I let bet the number of objects ...
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2answers
556 views

Poisson arrivals during an exponentially distributed interval

This is a marked homework question, so please try not to write complete solutions here: The number of customers that arrive at a service station during a time t is a Poisson random variable with ...
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1answer
34 views

Finding a tail-probability with the momentgenerating function

I wonder if it is possible to estimate $\mathbb{P}(X<t)$ with the moment generating function? This question popped up when I tried to proof this estimate $\mathbb{P}(T_a<t)\leq ...
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1answer
81 views

Quotes for statistical game

I have a simple scenario that displays quotes on a page. These quotes are numbers which players can choose, if certain events happen the player wins these quotes. Now the quotes for this scenario are ...
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88 views

Probability Distribution Of Sum of a Bernoulli and a Geometric Random Variable

Let X~Bernoulli$(\theta)$ and Y~Geometric$(\theta)$ where X and Y are independent. Let Z = X + Y. What is the probability function of Z? My thoughts are: Let Y be the number of failures until the ...
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1answer
178 views

Mathematical Statistics (Poisson Distribution)

The number of defects $Y$ per yard in a certain fabric has a Poisson distribution with parameter $\Lambda$. The parameter $\Lambda$ is itself a random variable with density ...
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155 views

Joint distribution of two marginal normal random variables

Question: Suppose we have: \begin{align*} \begin{bmatrix} X_1 \\ X_2 \end{bmatrix} \sim N\left(\begin{bmatrix} 6 \\ 3 \end{bmatrix}, \begin{bmatrix} 12 & 3 \\ 3 & 2 \end{bmatrix} \right) ...
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253 views

What's the mode of a bivariate Poisson distribution?

I have been looking at the bivariate Poisson distribution of the form as it is described at Wikipedia: http://en.wikipedia.org/wiki/Poisson_distribution#Bivariate_Poisson_distribution I was now ...
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116 views

About probability density function

I have a question about probability density function in my book. It reads: A Probability density function is of the form $p(x) = Ke^{-a|x|}$ , $x \in (-\infty,\infty)$.The value of $K$ is: 1] ...
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1answer
153 views

Finding the CDF of a random variable that has uniform distribution of outcomes.

I want to find the CDF of a a random variable $X(\omega) = e^\omega$, with the sample space $\Omega = [-1,1]$. The outcomes of $\Omega$ are uniformly distributed. What I've managed so far is to get ...
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1answer
210 views

Joint distribution by independent distributions

We have $N$ independent discrete finite random variables (RVs) $X_1,\dots,X_i,\dots,X_N$ where RV $X_i$ has $M_i$ finite number of elements. We are free to choose any distribution $f_i$ for RV $X_i$ ...
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147 views

Why do we need Triangle inequality for a probability distance measures

Kullback Leibler Divergence measure does not satisfy the triangle inequality. But the Hellinger Distance does. Do we really need triangle inequality for a probability distance measure. What does it ...
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205 views

Trying to work out the probabilities of a dice game I used to play

At college, my friends and I would sometimes waste time playing a game with dice. We would roll 25 dice, pick out all the dice that landed on a 6, then roll the rest. This would carry on until all the ...
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2answers
49 views

Random Variable with density and E(X)

$X$ is a random variable with value $0,1,2$ and $E(X)=1$, $E(X^{2})=3/2$. Find $f(x)$=the density of $X$ and find $E(X^{7})$=? Here is what I did: $E(X) = 0*f(0)+1*f(1)+2*f(2)$ $E(X^{2}) = ...
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37 views

Finding the distribution under a new measure

Suppose the the value of an stock is $S_t = S_{t-1}exp(\mu +\sigma Z_t) $ where $Z_t$ are standard normal variables. Find the distribution of ln($S_1/S_0$) under the Q measure given that dQ/dP is ...
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1answer
55 views

joint density and independence

if $(X,Y)$ is a random vector in $\mathbb{R}$ then $f_{X,Y}$ is said to be its joint density if $\mathbb{P}((X,Y) \in A) = \int_A f_{X,Y}(x,y) dxdy$ for all reasonable sets A. Now I have to show that ...
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1answer
810 views

a distribution of a sqrt of a Normal distribution

i have a Normal(0,1)=X. and (X_{1},....X_{20}). I have to calculate the distribution of $T=\sqrt{|Z|}$ with Z= $\dfrac{1}{20} \sum_{1}^{20}X_{i}$ and his average. I have done this, but Im not very ...
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82 views

Expected value of stock prices

Suppose the the value of an stock is $S_t = S_{t-1}exp(\mu +\sigma Z_t) $ where at time 0, $S = S_0$ and $Z_t$ are i.i.d standard normal random variables. Need to find the E($S_1$),E($S_2$), ...
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1answer
54 views

Finding the value of $f_{x+y}$ for multivariate normal distributon

Given that bivariate normal distribution is I need to find the value of $f_{X+Y}$ and the variable are standard normal. If the variables are standard normal, ...
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2answers
89 views

CDF of $X$ from joint CDF of $(X,Y)$

This question is from DeGroot's "Probability and Statistics"(Second Edition). Suppose that $X$ and $Y$ are random variables that can only take values in the interval $0\leq X\leq2$ and $0\leq ...
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1answer
133 views

How prove this distributions inequality $cov(\theta_{i},\theta_{j})\ge 0$?

Question: let random variable $\theta$ has dendity $f_{\phi}(\phi)$,and the random vector $\theta=(\theta_{1},\theta_{2},\cdots,\theta_{n})$,such $\theta_{i}|\phi$ are all independent from each ...
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55 views

find pdf from mgf for normal distribution.?

m.g.f of random variable X is equal to e^(t^2) for -infinity to infinity what is distribution of X? Here i suppose it is normal distribution but I am not able to slove.
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1answer
45 views

Got Stuck with these probability problems

I tried my best to solve 'em , but after waiting a few sheets of paper , I got nothing on me . A litle help from you guys might do the trick , Thanks ! ...
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43 views

Discrete Distribution Help

If p(n) = c(5/8)^n, 3 <= n <= infinity is a p.m.f. for a discrete random variable X, find (a) c, (b) the probability P(6 <= X <= 16), (c) the mean and (d) the variance Here's my work. I ...
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1answer
214 views

Joint Cumulative Distribution Function for $X\sim \exp( \lambda)$ and $Y=X^3$

Given $X\sim \exp(\lambda)$ and $Y = X^3$, what would be their joint cumulative distribution, $F(x, y)$? Since $X$ and $Y$ are dependent, I can't just integrate the product of their probability ...
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2answers
140 views

Probability mass function of closest integer of ratio of uniform random variables

Given two random numbers $X$ and $Y$ between 0 and 1 define $Z$ to be the closest integer to $X/Y$. I know what the ratio of two uniform random variables (that is I know its PDF and CDF). However, ...
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2answers
123 views

Random sum of random exponential variables

Let $t_1,t_2,\dots$ be independent exponential($\lambda$) random variables and let $Y$ be an independent random variable with $P(Y=n) = p(1-p)^{n-1}$. What is the distribution of the random sum $T = ...
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1answer
74 views

airplane weight

I need some help for my homework. On an airplane, 300 passengers will travel, whose weights are random and independent. The expectation of the total weight of all passengers is known to be 21000 kg. ...
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1answer
152 views

2 dimensional Brownian motion but not 3 dimensional Brownian motion

Let $W_t = (W_t^{(1)},W_t^{(2)},W_t^{(3)})$ be 3 dimensional Brownian motion. Let $X=sgn(W_1^{(1)})sgn(W_1^{(2)})sgn(W_1^{(3)})$. Define a 3 dimensional process $M_t$ as follows : $M_t^{(1)} = ...
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104 views

PMF : Determine the distribution function of X

The spectrum of a discrete random variable X consists of the points 1, 2, 3,..., n and its probability mass function (pmf) fi = P(X = i) is proportional to 1/i(i+1). Determine the distribution ...
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359 views

Moment-generating Function of a Continuous R.V. whose P.D.F is 1 from (0, 1)

I have been working on this problem for a few hours now and I feel I am missing something simple. The problem is to find the moment-generating function of a continuous random variable whose ...
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58 views

Deconvolution of distribution of diffraction reflexes

I'm a chemist stuck in a mathematical problem. Please bear with me as I'm trying to express myself in Math language. Let me explain in short terms the experimental method I'm using: X-ray ...
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1answer
114 views

Brownian motion at exponential time

I want to find the law of $B_T$, where $B_t$ is a brownian motion, $T$ is exp-distributed with parameter 1, with $B_t$ and $T$ being independent. My idea is to say that the density of $B_T$ is given ...
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67 views

Does this figure represent a cumulative distribution function?

Is this a c.d.f.? I have no problem for random variable $X$ at $-\infty<X<x_2$. But if p.d.f. were continuous in interval $x_2\leq X<\infty$ , then c.d.f. should have been continuous. If ...
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1answer
287 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 ...
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1answer
209 views

Minimum and maximum of random variable

Let $X$ be random variable such that $\begin{align} F_X(x) = 1- e^{-x} \end{align}$ if $x \ge 0$ and $F_X(x)=0$ in other case. Find distribution function $Y= \min(1,X)$, $Z=\max(1,X)$. If I have to ...
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1answer
75 views

If X and θ are both random variables and θ is the parameter of the distribution of X, are X and θ independent?

The answer appears to be no because the distribution of X is defined conditionally by θ which is also assumed to have a distribution as opposed to be a constant. Essentially, the formulation of the ...
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1answer
266 views

How to describe a “sum percentile”

I have values $w_1 \ge w_2 \ge ..\ge w_n$. I want to know the the highest possible threshold $w^{th}$ so that $$ \sum_{i:w_i>w^{th}} w_i \ge \alpha \sum_{i=1}^n w_i $$ where $\alpha \in [0,1]$. ...
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1answer
41 views

distribution of the product of a poisson and a bolzmann

What is the distribution of the product of two variables for which each of them has its own distribution(specifically one poisson and one bolzmann)? I found on wikipedia that for the sum of the two ...
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1answer
27 views

Probability distribution of halving a segment

Suppose you cut a unit segment ("a stick") in half. I'd like to find out theoretically the distribution of the length of, say, the left piece based purely on some plausible assumptions, such as ...
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1answer
35 views

How to sample N points between 0 and R if they are exponentially distributed?

The density of my points x $\in$ [0,R] is exponential: $\rho(x) \approx e^x$ How can I sample N points from there? Thanks,
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250 views

Probability question: given $P(A|B)$ and $P(B)$ how do I find $P(A)$?

I have a probability distribution for some quantity $A$ given a fixed $B$, i.e. $P(A|B)$. I also have a prior distribution $P(B)$ for $B$. I'm trying to find the distribution $P(A)$. I had thought ...
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53 views

Problem showing a limit ( involving the limit that equals the exponential function )

I am to show that if $(p_n)_{n \geq 1}$ is a sequence in (0,1) and $X_n\sim Bin(n,p_n)$ and $n\cdot p_n \to \lambda$ for $n\to \infty$ with $\lambda\geq0 $. Then $$ X_n(P)=\mu_n ...
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1answer
66 views

Probability Density and Distribution of a Sphere

I am given that the density function for the radii of a sphere is constant over 0 < r<5 and zero elsewhere and am asked for calculate the density function f(r) and the cumulative distribution ...
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1answer
28 views

Probability of orderings

Let $t_1,\ldots,t_n$ be a set of $n$ intervals, in the form $[l_i,u_i]$. I define the precedence $t_i \succ t_j$ as the event in which a sample drawn from $t_i$ is greater than a sample drawn from ...
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1answer
177 views

confidence interval of binomial disribution using standard deviation

Just as the normal distribution has the 68–95–99.7 rule with 68% of the data within +- 1 standard deviation and so on, does the binomial distribution too has something like that. Or does its being a ...
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2answers
138 views

Finding the conditional probability density using a joint PDF

I have a joint PDF $f_{X,Y}(x,y) = cxe^{-x}$ for $x > 0, |y| < x$ First, to determine $c$ I solved the double integral $$\int_0^\infty \int_{-x}^x cxe^{-x}dydx = 1$$ which gave me $c = 1/4$ ...
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59 views

$E[X^4]$ for binomial random variable

For a binomial random variable $X$ with parameters $n,p$, the expectations $E[X]$ and $E[X^2]$ are given be $np$ and $n(n-1)p^2+np$, respectively. What about $E[X^4]$? Is there a table where I can ...
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2answers
37 views

Finding the mass generating function of a continuous random variable given a pdf

The pdf is given, $$f_X(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-(x-\mu)^2}{2\sigma^2}}$$ Where $x\in(-\infty,\infty)$, $\sigma>0$, $\mu\in(-\infty,\infty)$. ...