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

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1answer
367 views

Binomial converges to Poisson

I want to prove that the Binomial distributions $X_n$ with $p_n$ probability converges in total variation norm to the poisson distribution $X$ if we have that $p_n \rightarrow 0$ and $np_n ...
1
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1answer
100 views

Finding efficiency of an estimator for Poisson random variables

$\newcommand{\eff}{\operatorname{eff}}$ I am asked to derive the efficiency of the estimator $\hat{\lambda}_1 = \frac{1}{2}(Y_1+Y_2)$ relative to $\hat{\lambda}_2=\bar{Y}$, where $Y_1,Y_2,\ldots,Y_n$ ...
1
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1answer
84 views

Expectation of a function in Poisson Distribution

Find the expectation of the function $\phi(x) = xe^{-x}$ in a Poisson distribution. My Attempt: If $\lambda$ be the mean of Poisson distribution, then expectation of $$\displaystyle \phi(x)=\sum_{x ...
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3answers
98 views

Chebyshev inequality for $n=1$?

Wikipedia suggests that Chebyhev's inequality is only true for $n \ge 2$, but I don't see why we have to exclude the case $n=1$? Is wikipedia right? Chebyshev
1
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0answers
76 views

What is the “true” entropy of a binary string?

Consider an infinite binary string $\sigma$ and define its entropy $$H_1 = -(p_0 \log_2 p_0 + p_1 \log_2 p_1)$$ with $p_i = \lim_{N\rightarrow \infty} N(i)/N$, $N(i)$ the number of $i$'s among the ...
1
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1answer
33 views

Finding the correct distribution model

The question is as follows: Let a be a positive constant and $X$ a random variable, so that: $P(X>x)=x^{-a}, x\ge1$ and 0 otherwise. Find the correct distribution model [i.e. uniform, ...
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2answers
32 views

Two dimensional Gaussian adding one dimensional Gaussian

Based on this question, from one dimensional Gaussian to two dimensional Gaussian, I have the following question. Any help is appreciated. Suppose $x$ is one-dimensional Gaussian distributed, with ...
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1answer
25 views

Probability distribution functions

If the probability density function is ($0\le x \le 1, 0\le y \le1$): (i) $f_{X}(x) = \frac{3x^{2}}{2} + x$ (ii) $f_{Y}(y) = \frac{3y^{2}}{2} + y$ Find the distribution functions $F_{X}(x) = P(X\le ...
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1answer
52 views

Binomial Distribution; Using a mean to suggest probability of success.

and thanks for taking the time to look at my question. This was for a homework task, on binomial distribution, which i thought i understood. A consignment of china mugs is packed for ...
0
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1answer
104 views

Help needed to derive combinatorics formula.

I am having troubles understanding a combinatorics formula. I would appreciate any ideas or hints, leading to an explanation how this formula might be derived. I came across the formula reading a book ...
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2answers
79 views

Is it reasonable to calculate $P(X_1 = X_2)$ given that $X_1$ and $X_2$ are continuous random variables?

To be specific, suppose $X_1$ and $X_2$ are independent exponential random variables with parameters $\lambda_1$ and $\lambda_2$; what is $P(X_1 = X_2)$? According to section 5.2.3 of the book ...
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1answer
166 views

How do you calculate a binomial distribution with k > R as opposed to k = R

I'm given the formula: $\displaystyle P(X = k; n, p) = \binom {n}{k} * p^k * q^{n-k}$ And we need to work out the binomial coefficient by hand, instead of using C(n,r). So I have a question: "Some ...
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1answer
55 views

Question on integrating probability

Let $X,Y$ be two independent random variables with distribution functions $F_X,F_Y$. I need to show that $$P(Y\leq y+X)=\int F_Y(y+x)\mathrm{d}F_X(x)$$ I have no idea where to start.
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1answer
28 views

Distribution of $U + Y$ $\mod 1$

Let $U$ ~ Unif$[0,1]$ be uniformly distributed on $[0,1]$. Let $Y$ be some random variable, independent from $U$. What is the distribution of the random variable defined as $X=U+Y \mod 1$? Can ...
0
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1answer
30 views

Dealing with this random variable problem

$X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\infty)$. Also $t\in(0,1)$. $f_1>0$ is a positive constant and $f_2,f_3,f_4\ldots$ are positive functions of one variable. ...
5
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3answers
243 views

Expected value of number of draws

We have $5$ number in a bag: $(1,3,5,7,9)$. We draw one from the bag and then put it back. We do this until the sum of the numbers can be divided by $3$. Whats the expected value of the number of ...
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2answers
3k views

Waiting time in a bus stop

I have a question in a Probability theory and think that it could be solved by exponential distribution. However I'm not confident for dealing with this. Hope to get some helps. Thanks in advance. A ...
2
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1answer
28 views

Broken trucks at a road

If three trucks break at locals random distributed of a road with lenght $L$, find the probability that $2$ of those trucks are not at a greater distance than $d$, fot $d \leq \frac{L}{2}$ My ...
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0answers
53 views

Expected number of errors in a magazine page

The expected number of typographic errors in a page of a certain magazine is 2.What is the probability that an article of 10 pages has 2 errors? My attempt: I thought that if $X$ is the r.v that ...
0
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1answer
282 views

Poisson distribution, (conditional) probability question

Suppose Bob receives on average one call per night from his father. Find the probability that 7th January was the third night this year (starting on 1st January) when no night calls were received. I ...
2
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1answer
116 views

Probability with same number of heads and tails

I have this question that I don't know where to begin, any help will be greatly appreciated. Consider the following random experiment. Toss a coin until the same number of heads and tails have been ...
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1answer
20 views

Normal Distribution how $N(x-x_n|0,\sigma^2) = N(x |x_n,\sigma^2) $

I read an expression Could someone explain the step $N(t-t_n|0,\sigma^2) = N(t | t_n,\sigma^2) $ ?
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0answers
38 views

PDF describing nth term in continued fraction

For a real number r chosen uniformly at random in the range (0,1), what's the marginal Probability Density Function that describes the nth term in the continued fraction representation of r? What ...
0
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1answer
60 views

Find moment generating function given rth moment without Laplace

Let W be a random variable. Given the rth moment, $E(W^{r}) = \frac{r!+6^{r}}{2^{r}} = M_w^{(r)}(0)$, $r \geq 0$, how does one derive $f(w)$? I know that $M_w(t)=E(e^{wt}) = \begin{cases} ...
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1answer
128 views

How to obtain k and λ of this Erlang distribution?

I have a dataset that represents the difference, in milliseconds, between an input event (a key pressed in a keyboard) and the next one. I have grouped the data in sets of 25ms, e.g. (0, 25], (25, ...
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1answer
74 views

Poisson distribution

Consider this problem: The number of telephone calls made to an exchange is Poisson distributed with a mean of 6 calls per hour. Find the probability that a call will be made within 10 minutes given ...
0
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1answer
19 views

Distribution function of the random variable $R_2=e^{-R_1}$

An absolutely random variable $R_1$ is uniformly distributed betweem $-1$ and $+1$, find the density and the distribution function of the random variable $R_2$, where $R_2=e^{-R_1}$. $R_1$ is ...
0
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0answers
333 views

How can I derive the PDF from conditional probabilities?

I have some function $P(i)$ which is the probability of success for an experiment on the $i$th trial. The probability mass function for the first successful trial is: $$PMF(n) = \left( ...
3
votes
2answers
283 views

Sum of independent Poisson random variables is a Poisson random variable

Suppose $x_1$ and $x_2$ are independent Poisson random variables with parameters equal to $\lambda_1$ and $\lambda_2$ respectively. Show the sum of $x_1$ and $x_2$ is also a Poisson random variable ...
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0answers
231 views

Solving Probabilities for M/M/1 Queue Waiting Time Generating Function

I "believe" that generator, $\bf Q$, of the waiting time distribution for the $M/M/1$ queue is given by the following (I'm not 100% sure if this is even correct): $\bf Q$ = $\left( ...
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0answers
29 views

unimodality and continuous

i would like to ask question about unimodality of probability function ,from wikipedia http://en.wikipedia.org/wiki/Unimodal it says that In mathematics, unimodality means possessing a unique mode. ...
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3answers
60 views

Question about calculating at Uniform distribution

A train come to the station $X$ minuets after 9:00, $X\sim U(0,30)$. The train stay at the station for 5 minutes and then leave. A person reaches to the station at 9:20. Addition: There was no train ...
2
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2answers
79 views

How to prove something at Uniform distribution…

$X\sim U (0,1)$. The point $X$ divides $[0,1]$ to two parts. $Y=\frac{\text{The big part}}{\text{The small part}}$. ($Y$ is the ratio... $Y\ge1$). What is the density function of $Y$? I'd like to ...
0
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1answer
40 views

Solving this random variable problem

This is an earlier problem Proving this random variable problem but generalised, maybe you want to take a look at that one first? $X_1,X_2,X_3,\ldots$ are IID random variable taking values in ...
1
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2answers
59 views

Proving this random variable problem

$X_1,X_2,X_3,\ldots$ are IID random variable taking values in $(-1,\infty)$. Also $t\in(0,1)$. Define random variables $Y_1,Y_2,Y_3,\ldots$ recursively like $$Y_1 = (1+tX_1)$$ $$Y_n = ...
1
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1answer
53 views

Uniform distribtion: clarification of $f_X(x)$

I have $Y=2(X-1)^2 -1$ where $X$ is uniform distributed on $[0,2]$ I want to find the pdf of $Y$ and expected value of $Y$. My question is just: Does $X$ have pdf $f_X(x)= \frac{1}{2}$?
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1answer
37 views

Expectation of Continuous variable.

Given the probability density function $$ f(x) = \begin{cases} \frac{cx}{3}, & 0 \leq x < 3, \\ c, & 3 \leq x \leq 4, \\ 0 & \text{ otherwise} \end{cases} $$ I have found $c$ to be ...
0
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2answers
47 views

Chi Square Test for one variable

I got a question about the use of Chi Square test. Let's assume I am conducting a survey. And I have a question: "Have you ever heard of the Internet"? The possible answers are: "Yes", "No", "Not ...
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2answers
83 views

probability distribution of a random selection from one of two bernoulli random variables

Say I have two bernoulli random variables, $X$ and $Y$, and that I want to randomly select from either one of them with some probability $p$. In other words: $X$ ~ $Be(p_X)$ $Y$ ~ $Be(p_Y)$ Then I ...
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0answers
48 views

What is the expectation of $X^2$ where X has a truncated normal distribution?

Suppose that $X\sim N\left(a,\mbox{ }\sigma^{2}\right)$, what is $E\left\{ \left[1\left\{ X>b\right\} \exp\left(X\right)\right]^{2}\right\}$? $b$ is a constant.
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1answer
236 views

P.d.f. of $XY$, where $X, Y$ are independent uniformly distributed over $[0,1]$ [duplicate]

I tried to change the variables: Let $U=XY$ and $V=Y$; so then the Jacobian is $1/v$. So joint pdf $g(u,v) = f(x,y)\cdot (1/v) = 1/v$ Would you then integrate over $v$ from $0$ to $1$ to get the ...
0
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1answer
55 views

Limit of Poisson Distribution

Just for fun, I'm looking at the concentration of the Poisson Distribution near it's mean. For $\lambda=10$, there is a 36% probability of being within 10% of the mean. For $\lambda=100$, that ...
1
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1answer
33 views

Simple distribution problem

I have $F_X(x)= \frac{1}{2} + \frac{1}{\pi}arctan(x)$ And I know $Y = aX+b$ Does that make $F_Y(y) = \frac{1}{2} + \frac{1}{\pi}arctan(\frac{y-b}{a})$ Pretty sure $F_Y(y) = F_X(g^{-1}(x)$ where we ...
3
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2answers
1k views

How to find nth moment?

I'm quite new to the field so please bare with me. Problem: Let ξ be a random variable distributed according to a log-normal distribution with parameters μ and $σ^2$, i.e. log(ξ) is normally ...
2
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0answers
136 views

Hitting time of a maximum of random walk converges to that of Brownian motion

Suppose $S_n$ is a simple random walk; formally, $S_n=\sum_{i=1}^n X_i$ for $X_i\sim\mathcal{U}(-1,1)$, i.i.d.. Denote by $M_n$ the maximum of the random walk on $n$ steps; formally, $M_n=\max_{0\le ...
1
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1answer
70 views

Which Queue to Join at the Super Market

Last night I started wonder about the fastest way to take a shopping trip with my university flat mates and was wonder about how we should queue for the check out. I have a feeling that queue theory ...
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0answers
32 views

Maximum possible distance between two vectors sampled from n-variate Gaussian

What would be the probability distribution of the distance between two vectors sampled from n-variate Gaussian distribution? Thanks.
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5answers
287 views

Find the distribution of $X_1^2 + X_2^2$? [duplicate]

Let $X_1$ and $X_2$ are independent $N(0, \sigma^2)$ which means (mean = 0, variance = $\sigma^2$) random variables. What is the distribution of $X_1^2 + X_2^2$? My approach is that $X_1\sim N(0, ...
0
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1answer
19 views

Is the statement: $p\left(\left.y\right|h^{-1}\left(\varphi\right)\right)=p\left(\left.y\right|\varphi\right)$ correct?

Say I have a likelihood function $p\left(\left.y\right|\theta\right)$ and I make the reparameterization $\varphi=h\left(\theta\right)$ using the bijective function $h$ with inverse $h^{-1}$. Then it ...
1
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1answer
236 views

Is a log-normal distribution uniquely determined by its moments or not?

Wikipedia states that A log-normal distribution is not uniquely determined by its moments $\text{E}[X^k]$ for $k\ge 1$, that is, there exists some other distribution with the same moments for all ...