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

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2answers
46 views

Expected value (and distribution) of sum of six balls labeled 1-49, no replacement.

The problem stems from the Spanish lottery, in which 6 balls are drawn from an urn with 49 balls, labeled 1-49, without replacement. My goal is to figure out the expected value of their sum, and if it ...
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0answers
19 views

Joint p.d.f $Y=x_1/x_2$ for two independent continuous random variables $X_1$ and $X_2$

The question reads like this: Two independent continuous random variables $X_1$ and $X_2$ have a joint p.d.f $f(x_1,x_2)$. Determine the p.d.f of $Y=X_2/X_1$, assuming $Y>0$. (That is $Y$ is ...
2
votes
1answer
111 views

Hypothesis test between two normal distributions

Let $T_1,T_2,\ldots ,T_ n$ be i.i.d. observations, each drawn from a common normal distribution with mean zero. With probability $1/2$ this normal distribution has variance $1$, and with probability ...
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1answer
67 views

Trajectory estimation

The vertical coordinate (“height") of an object in free fall is described by an equation of the form $x(t) = \theta _0 + \theta _1t + \theta _2 t^2,$ We assume that $\theta_0$ is a known constant. We ...
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0answers
18 views

Error in cumulative distribution function due to error in mean

Suppose that $x_1,x_2,x_3$ are three random variables with mean $\mu_1=5, \mu_2=3, \mu_3=1$, respectively and each with variance $\sigma = 0.1$. The joint cumulative distribution function(cdf) of ...
1
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1answer
23 views

Calculating expected value of a pareto distribution

Suppose that you have a Pareto product distribution function defined by: $$ f(x;k;\theta)= \begin{cases} \frac{k\theta^k}{x^{k+1}} & x \ge\theta \\ 0 & x \lt \theta \end{cases} $$ How would ...
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0answers
18 views

probability problem using Chebyshev's inequality

Suppose that a die has its "3" side changed to a "2". The problem is to first find a lower bound on the probability $P[3\leq X \leq 4]$ using Chebyshev's inequality. Then if we roll the die $n$ ...
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4answers
9k views

Difference between power law distribution and exponential decay

This is probably a silly one, I've read in Wikipedia about power law and exponential decay. I really don't see any difference between them. For example, if I have a histogram or a plot that looks like ...
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1answer
46 views

Are $X$ and $X+Y$ independent, if $X$ and $Y$ are independent? [closed]

As asked in the title? Does the independence of two random variables $X$ and $Y$ imply the independence of $X$ and $X+Y$? If so, what's the easiest way to prove that?
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2answers
39 views

A problem in joint random variable (bivariate normal)

Suppose that $Y_{1}$ and $Y_{2}$ follow a bivariate normal distribution with parameters $\mu_{Y_{1}}=\mu_{Y_{2}}=0$, $\sigma^{2}_{Y_{1}}=1, \sigma^{2}_{Y_{2}}=2$, and $\rho=1/\sqrt{2}$. Find a linear ...
1
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1answer
34 views

Bias of $\sigma^2$ estimator

I need to find the bias of $\frac{\sum(x_{i}-\bar{x})^2}{n+1}$ for $\sigma^2$. To do so, one must take its expectation but add and minus $\mu$ from the summation part so we can bring $\sigma^2$ into ...
1
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2answers
44 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 ...
0
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1answer
22 views

Calculate the Probability of a Normally Distributed Random Sample

Please i would like to understand these problems about probability distributions, I can't find a right solution for this problem. I have a variable X which is the level of glucose in blood and is ...
1
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2answers
67 views

Bivariate normal density function [closed]

$W$ and $Z$ have the bivariate normal density function $$f(w,z)=\frac{1}{2\pi\sqrt{1-\rho^2}}\exp\left\{-\frac{1}{2(1-\rho^2)}(w^2-2\rho wz+z^2)\right\}$$ for $w,z\in\mathbb{R}$ and fixed ...
0
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0answers
27 views

Finding Prob Distribution Functions

I am trying to find the PDF of certain variables and I am a little stuck. Here are the questions: Particles have radius $X$ distributed with density $f_X(x)=\frac{25}{12x^3}$ between $[1,5]$. The ...
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3answers
49 views

Combinatorial Distribution with random sample I believe

I have no idea what kind of distribution this is and that is what I would like. Balls are numbered 1 to $N$. We select a sample of $n$ at random. Let $Y$ be the largest number in the sample. Find ...
2
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1answer
28 views

New characteristic functions from old

I am doing an exercise which says: If $f$ is a characteristic function, then show that $$ F(t):= \int_0^{\infty} f(ut)e^{-u}du $$ is again a characteristic function. Is this answer correct? Let ...
0
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1answer
95 views

Fair coin flipped twice independently

Consider a fair coin that has 0 on one side and 1 on the other side. We flip this coin, independently, twice. Define the following random variables: X = the result of the first coin flip Y = the sum ...
2
votes
2answers
32 views

Pointwise convergence and L1 convergence in bounded mass case

I have a question regarding convergence modes and their relationships, my problem is actually an application to probability, How to prove that : ${f_n}$ and $g$ are probability density functions such ...
1
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1answer
19 views

If $x$ is a $\chi^2_{N-n}$ RV. what is $x/N$ as N goes to infinity

We know that if we have $N-n$ gaussian iid RVs $\{e_i\}$ with mean $0$ and variance $1$, the RV $x = \sum e_i^2$ is $\chi^2$ distributed with $N-n$ degrees of freedom. We have $N$ larger than $n$. I ...
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0answers
14 views

probability distribution of 2nd-order vector autoregressive models

Suppose I have a 2nd-order vector autoregressive model VAR(2): $$ \mathbf{y}_t=\mathbf{\Phi_1}\mathbf{y}_{t-1}+\mathbf{\Phi_2}\mathbf{y}_{t-2}+\mathbf{\varepsilon}_t $$ where ...
0
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0answers
31 views

What is the maximum entropy distribution over all integers (ie. including negative ones) with fixed mean and variance?

I know that the maximum entropy distribution with over the non-negative integers fixed mean is a geometric distributions. However, I cannot find conclusive information about what are the maximum ...
0
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0answers
17 views

Cumulative beta density calculation. [duplicate]

The beta distribution of y, w.r.t $\alpha,\beta, min - a \text{ and } max - c $ is. $$f(y; \alpha, \beta, a, c) = \frac{ (y-a)^{\alpha-1} (c-y)^{\beta-1} }{(c-a)^{\alpha+\beta-1}B(\alpha, \beta)}$$ ...
2
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0answers
28 views

Marginal Distribution of the sum of Bernoulli rv

consider the conditional on probabilities $p_1, \ldots, p_n$, with independent Bernoulli random variables $Y_1, ..., Y_n$ given that $P(Y_i = 1\mid p_1, \ldots, p_n) = p_i, \ P(Y_i = 0\mid p_1, ...
1
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2answers
29 views

every pdf can be regarded as a marginal distribution of a joint pdf

suppose we have functions $g\ge0,h\ge0$ that $\int g \, dx=1 , \int h \, dy = 1$. it means $g$ is pdf for random variable $X$ and $h$ is pdf for $Y$. now how we can prove that there is function ...
2
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2answers
91 views

Proving Brownian Motion has Stationary Increments

In Oksendal's 'Stochastic Differential Equations', we define Brownian Motion as follows: Fix $x\in\mathbb{R}^n$ and define for $y\in\mathbb{R}^n$: $$p(t,x,y)=(2\pi ...
0
votes
2answers
61 views

Difference between two real roots with uniformly distributed coefficents

I have a question that first I need to know what is happening, but then I also need to code it in a program called APPL, which is an extension from Maple18 that I really have never used, yet I have ...
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0answers
9 views

Transforming $\frac{\chi^2_{v_1}}{\chi^2_{v_2}}$ into an F- distribution

my first post here, so I hope I did it in the correct place and with the appropriate tags and all. I have two Random Samples: $$ Y_1 ... Y_n \text{ , where } Y_i=(1/\theta)e^{-y_i/\theta} $$ and ...
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1answer
20 views

variance of multivariate normal

currently trying to compute the first two moments of the multivariate distribution. Got an extremely helpful answer to show that $\mathbb{E}[x]=\mathbb{m}$, with $x \sim ...
1
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0answers
22 views

Hammersley–Chapman–Robbins bound for Rice distribution

I am trying to evaluate the Hammersley–Chapman–Robbins bound for the variance of an unbiased estimate $\hat{\alpha}$ of $\alpha$ (for a given $\sigma$) for the Rice distribution: $$p(x|\alpha,\sigma) ...
2
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1answer
43 views

Negative Binomial

Let $X$ be a negative binomial with parameters $r$ and $p$. So $X$ is the number of trials $k$ till the $r^{th}$ success. My first question is determine which values of $k$ the ratio ...
1
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1answer
458 views

How to prove that the binomial distribution is approximately close to the normal distribution when $np(1-p) \geq 10$

I would like a formal proof for this "rule of thumb." Can you assist me in getting to this solution? I require the insights and creativities of mathematicians. We know that if $np(1-p) \geq 10$ the ...
1
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1answer
25 views

Difference of Exponential Random Variables / Linear Transformations of RVs

Suppose X and Y are both distributed exponentially with parameter $\lambda$ and $\mu$ respectively. I am trying to find the distribution of X - Y via this method and it does not seem to be working, ...
0
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1answer
26 views

Independent variable vs. Uncorrelated variable confusion. How do I interpret this?

I'm reading Time Series Analysis and Forecasting by Example by Søren Bisgaard and Murat Kulahci and I'm having trouble conceptualizing a particular passage and it's bugging me enough that I can't move ...
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2answers
24 views

Probability and series. (I just need help with the arguments provided )

I have this problem: Suppose that random variable X have possible values $1,2,\ldots,$ and $P(X=j)=\dfrac{1}{2^j}$. Calculate P(X is even). I did the following; I noted that $P(X=2k)=\dfrac{1}{4^k}$ ...
0
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1answer
32 views

From a stack that contains 25 articles, 5 of them are defective, then we choose 4 at random.

I have the following problem; From a stack that contains 25 articles, 5 of them are defective (20 are not), then we choose 4 at random. Let X be the number of defective articles founded. Get the ...
1
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1answer
29 views

Poisson and Cumulative Distribution Double Check

this is simply me double checking my answer. Let Y be number of fish caught on a trip with Poisson distribution and $\lambda=4$. What is prob of catching 3 or fewer trout on trip? I said this was ...
2
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1answer
34 views

Normal Distribution Probability with known mean and variance

I believe I am quite close to solving this, but I would just like to double check some of these answers. Two species have different size toes. Lengths of toes of species X is normal distributed with ...
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1answer
40 views

Distribution of sum of $m$ independent random variables

Let $A_m$ be the sum of $m$ identically distributed random variables that are independent and that have an exponential distribution with parameter $\mu$. How do I prove that $A_m$ has a gamma ...
3
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1answer
22 views

distance function between histograms accounting for bucket distance

I'm trying to find a good distance measure for histograms that has the following properties: if histogram A and B have high values in buckets further apart, they're more different than if they had ...
0
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1answer
35 views

Finding density function, plus showing $X \sim F$ where F is cdf of X, $X = F^{-1}(U)$, $U\sim unif(0,1)$ [duplicate]

Suppose $X$ has a continuous, strictly increasing cdf $F$. Let $Y = F(X)$. What is the density of $Y$? Then let $U \sim unif(0,1)$ and let $X = F^{-1}(U)$. Show that $X\sim F$. The first part seems ...
0
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1answer
343 views

How do you transform Gamma to Chi-squared distribution

Here is the question not sure how to turn a Gamma into a Chi-Squared: Suppose $X_1....X_n$ is a sample from the distribution Gamma($\alpha=3,\ \lambda=\theta$) with unknown $\theta > 0$. We wish ...
2
votes
1answer
21 views

Conditional Probability with Ordered Stats

I am very close to solving this, but this last part is killing me on how to solve it. I dont really know where to begin. Scores run from 0 to 5 with density $f(x)=c(x^2-6x+10)$. An intermediate score ...
1
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1answer
33 views

Two questions about Almost sure convergence and Uniform integrability

Let $X_n$ and $Y_n$ be two sequences of random variables such that $X_n\stackrel{n}{\rightarrow}C$ almost sure and $Y_n\stackrel{n}{\rightarrow}C$ almost sure, $C\in \mathbb{R}$. Suppose that $X_n$ ...
1
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2answers
376 views

What is the probability that a student knows the answer given that he has answered it correctly,…?

A large class in stochastic processes at at a school is taking a multiple choice test. For one particular question with m proposed multiple choice answers, the fraction of students who know the answer ...
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1answer
16 views

Use the tail bound to estimate the probability

The heights of trees in a particular forest follow a normal distribution with mean 60 feet and standard deviation 10 feet. The tail bound for the standard normal distribution (i.e. X ∼ N(0, 1)) is: ...
1
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1answer
88 views

What can we say about $1-F(x) = x$?

$F(x)$ is a probability distribution. Is there any useful characterization of the solution to: $1-F(x) = x$? More specifically, can we say anything about the solution in terms of it's relationship ...
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1answer
21 views

multivariate normal moment derivation

I am having trouble deriving the mean for a multivariate normal for $\mathbf{x} \sim \mathbb{N}(\mathbf{m},\Sigma)$: $$ \mathbb{E}[\mathbf{x}]= \int_{R^d} \mathbf{x} ...
0
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1answer
32 views

Prove that an absolutely continuous cdf is continuous

Let $F(x_1,\ldots,x_d)$ be an absolutely continuous distribution function. How to prove that $F$ is continuous? Thank you.
1
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1answer
32 views

Poisson distribution

Suppose that in a population the probability of survival of an individual would survive is 99/100. If 5 people in the population were to buy an insurance, Using Poisson distribution what is the ...