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

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1answer
41 views

Find the probability density function of $Y=X+Y$

If the joint density of $X_1$ and $X_2$ is $$f(x_1,x_2) = 6e^{-3x_1-2x_2}, x_1 >0; x_2>0$$ Find the probability density function of $Y=X_1+X_2$ We did this as a class example but never ...
0
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2answers
37 views

Is XY the joint distribution over X and Y?

The definition of covariance is $Cov(X,Y) = E[XY] - E[X]E[Y]$ I can't wrap my head around what $XY$ is supposed to be. I suspected it to be the joint distribution over $X$ and $Y$ but I could not ...
0
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0answers
79 views

What is the premium such that it is equal to the $90^{th}$ percentile of the distribution of total claims?

A company has a one-year group life policy that divides its employees into two classes as follows: Class, Probability of Death, Benefit, Number in Class, A, 0.01, $...
-2
votes
1answer
86 views

Very hard probability problem [closed]

One of the first 6 positive integers is to be chosen by casting an unbiased die. Let this random experiment be repeated five independent times. Let $X_1$ be the random variable representing the number ...
0
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0answers
31 views

distribution of infinite sum of independent but non-identical normal variables

For $i=1,2,\ldots,n$, suppose $X_i \sim N(0,\Omega_{i})$, where $\Omega_{i}$ is of dimension $k\times k$. It is known that $\frac{1}{\sqrt{n}} \sum_{i=1}^{n} X_i \sim N(0, \overline{\Omega})$, where ...
1
vote
3answers
57 views

Finding maximum of two variables

Given $X$ is uniform on $[0, 10]$. Let $$Y = \max(5, X).$$ Determine Var(Y). I'm familiar with how to find the variance of a uniform random variable, as well as the max of two random variables. ...
0
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0answers
19 views

what do we get if we compare fisher information of two distributions?

What do we get if we compare fisher information of two distributions belonging to different family of distributions with shape and scale parameters say we compare it for generalised exponential and ...
4
votes
1answer
123 views

Solution to a certain moment problem

I'm looking for a function $f$ that satisfies $f(x)\geq0$ $\int f(x) \mathrm{d}x=1$ $\int xf(x) \mathrm{d}x=0$ $\int x^2f(x)\mathrm{d}x=1$ $\int x^4f(x)\mathrm{d}x=\delta$ $\int x^5f(x)\mathrm{d}x=\...
0
votes
2answers
29 views

Finding standard deviation given joint probabilities

I'm trying to find the standard deviation of $Z = X + Y$ given the following table: I'm getting $E[Z^2] = 36.31$ and $E[Z] = 5.45$, giving me a variance of $Var[X] = 36.31 - (5.45)^2 = 6.6075$ ...
0
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0answers
11 views

$\exists \mathcal{A},\mathcal{B}:X\sim \mathcal{A}\Rightarrow \frac{p}{\sqrt{q+rX}}\sim \mathcal{B}$?

Does there exist a parametric distribution $\mathcal{A}$, such that: $X\sim \mathcal{A}\Rightarrow\frac{p}{\sqrt{q+rX}}\sim \mathcal{B}$ for some parametric distribution $\mathcal{B}$ Where $p,q,r&...
1
vote
1answer
27 views

A die is cast $3$ independent times, let $Y = \max(X_1,X_2,X_3,)$…

A die is cast $3$ independent times. Let $X_i$ be the random variable representing the number on the face appearing at the $i$th cast. Let $Y$ be the random variable defined by $Y = \max(X_1,X_2,X_3)$....
0
votes
1answer
39 views

How does sampling affect the distribution of frequencies of individual types?

Consider a population of size N in which individuals can be of x different types. Take a sample (with replacement) of size ...
0
votes
3answers
65 views

Probability of a Poisson random variable taking a specific value [closed]

Let $X$ be a random variable with Poisson distribution with mean $\lambda=9$. Knowing that $P(X=8)=0.131756$, compute $P(X=9)$. How do I approach this problem?
3
votes
1answer
79 views

Probability of achieving Maximum Value in a Probability Density Function

As part of my (incomplete) lecture notes I've been provided with this example but, alas, with no supplemented solution. As this is the first (& only) example following on from Theorem 1.4 (...
0
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0answers
26 views

Minimax of Negative Binomial

Studying for the quals and I am in a deep confusion. This is an exercise from Mathematical Statistics: A Decision Theoretic Approach by Thomas S. Ferguson. Let $X$ has density $$f(x|\theta)={r+x-1 ...
2
votes
2answers
77 views

If X is log-normal, is: $\frac{a}{\sqrt{b+cX}}$?

I am working for the first time with log-normal distributions and I want to verify whether the following statement is true. I am not sure whether all the properties of the log-normal distribution hold ...
-1
votes
1answer
47 views

random distribution/probability

Let $a$ and $b$ be positive integers with $a \le b$, and let $X$ be a random variable that takes as values, with equal probability, the powers of $2$ in the interval $[2^a, 2^b]$. The question is ...
0
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0answers
29 views

probabilistic rbotics

we will apply Bayes rule to Gaussians. Suppose we are a mobile robot who lives on a long straight road. Our location x will simply be the position along this road. Now suppose that initially, we ...
0
votes
1answer
54 views

Deriving Mean for Negative Binomial Distribution.

A negative binomial distribution is given by : $P(X=x)\:: \binom{x-1}{k-1}p^{k}(1-p)^{(x-k)}$ , where p is probability of a success. where , x = k , k+1 , k+2 ,..... and so on. Mean is given by : $...
4
votes
2answers
92 views

Lower bound for (function of) density of well-behaved random variable

Suppose we have a non-negative random variable $\tilde{\theta}$ such that $\mathbb{E}\tilde{\theta} = a > 0$, with finite variance $\sigma^2$. $\tilde{\theta}$ can take on values from $0$ to $\...
0
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1answer
60 views

What does statistical significance literally mean? [closed]

I am getting confused when trying to determine the statistical significance of my results. For instance, I was attempting to make a conclusion of reliability of a model after 100000 trials depending ...
-1
votes
1answer
63 views

Problem with Binomial distribution

Can you help me with this problem - how to solve it or what methods to use? The amount of burnt lamps in a device is a binomially distributed random variable $$X \sim \mathcal{Bin}(3, 1/3)$$ The ...
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2answers
46 views

Does logarithm of Gaussian image still gaussian distribution?

I have an image 2D that pixel intensity follows multi Gaussian distribution such as $$p \left( I(x) \in \Omega_i \mid (I(x)\right)=\frac{1}{2\pi \sigma_i}\exp\left(-\frac {(I(x)-\mu_i)^2}{2\sigma_i^...
0
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1answer
34 views

Find $P(796.2 \leq \sum_{i=1}^{16}(X_i-50)^2 \leq 2630)$ with a sample of $n=16$ and $X \sim N(50,100)$

If $X_1,X_2, ..., X_{16}$ is a random sample of size $n=16$ from the Normal Distribution $N(50,100)$, determine: $$P(796.2 \leq \sum_{i=1}^{16}(X_i-50)^2 \leq 2630)$$ Okay well I know that $\...
1
vote
2answers
75 views

Uniqueness of moments for probability distributions with infinite moments.

I was taught the collection of a distribution's moments uniquely defined the distribution. Recently, I have been studying Pareto distributions, which have infinite means for shape parameters less than ...
0
votes
1answer
124 views

Probability that 5 out of 7 bulbs will produce white flowers

If you have a bag with 25 tulip bulbs that will grow into white, yellow or red flowers. You want to plant 7 bulbs. Each bulb in the bag, independently of the others, grows into a white tulip ...
2
votes
1answer
81 views

stochastic dominance definition

I was wondering if, for positive random variables $X$ and $Y$, $\Pr(X\geq Y)\geq 1/2$ implies $\Pr(X\geq x)\geq \Pr(Y\geq x)$. Intuitively it "makes sense", since $X$ tends to be more often bigger ...
0
votes
1answer
52 views

$\chi^2$ distribution Stoch. increasing in non-centrality parameter

i.e for fixed $\nu>0$ if we have $\gamma_2 > \gamma_1>0$ then $\chi^{2}_{\nu}(\gamma_2)\succeq\chi^2_{\nu}(\gamma_1)$ where '$\succeq$' denotes stochastically larger. The convention that I ...
4
votes
1answer
92 views

How long before the prey can escape?

I've (sort of) come across the following problem in my research. The actual scenario is a little abstract to explain, so I'm rephrasing the problem in terms of a predator/prey scenario. I'm tagging ...
1
vote
1answer
42 views

$\frac{\chi^2_n}{n}$ Stochastically increasing in $n$?

I was wondering whether $\frac{\chi^2_n}{n}$ is stochastically increasing in $n$. My main problem: Suppose $\hspace{5pt}\frac{(n-p)\hat{\sigma}^2}{\sigma^2} \sim \chi^2_{n-p}$. Then the expected ...
0
votes
1answer
47 views

Given the moment generating function of a continuous-type r.v, how to find the p.d.f?

Say for $t<1$: $$M(t) = \frac{1}{(1-t)^2}$$ How to find the p.d.f of the random variable? $$M(t) = E(e^{tx})=\int_{-\infty}^{+\infty}e^{tx}f(x)dx$$ How do we find: $f(x) = xe^{-x}$ on $(0,+\...
1
vote
1answer
154 views

Proving that the variance is non-negative

$$(E(X))^2 = \left( \int_{-\infty}^{+\infty}xf(x)dx \right)^2 \le \int_{-\infty}^{+\infty}x^2(f(x))^2dx \le \int_{-\infty}^{+\infty}x^2f(x)dx = E(X^2)$$ Because of cauchy-schwarz inequality and $f(x) ...
2
votes
3answers
75 views

Distribution of sum of random variables

Let $X_1, X_2, . . .$ be independent exponential random variables with mean $1/\mu$ and let $N$ be a discrete random variable with $P(N = k) = (1 − p)p^{k-1}$ for $k = 1, 2, . . . $ where $0 ≤ p < ...
0
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2answers
45 views

Neighbor Interaction in a Random List

Assuming a random arbitrarily long list where each element has a $50\%$ chance of being a $0$ or a $1$, such as: $0001101101$ What is the chance of having a neighbor that isn't the same? For ...
0
votes
1answer
35 views

Can the independence of random variables hold for their functions?

Suppose $X$ and $Y$ are two independent continuous random variables on $\mathbb{R}$. Define: $f:\mathbb{R}\mapsto\mathbb{R}$ as a $C^\infty$ map on $\mathbb{R}$. Then is it possible to find the ...
2
votes
1answer
86 views

How to find the density of $Y=g(X)$ in this case?

I have a vector $X=(1,X_2,X_3)$, where $(X_2,X_3)$ is a random vector in $\mathbb{R}^2$. Now consider $Y=g(X)=X/\|X\|$. What is a density function of $Y$ with respect to the uniform spherical measure,...
0
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0answers
21 views

Approximate CDF of arbitrarily aggregated random variable

I would like to know if my solution for the following is mathematically correct in general: I have a random variable $Z$ that is an arbitrary function of two other rvs $X$ and $Y$, so: $Z = f_{arb}(X, ...
0
votes
1answer
68 views

Variance of Transformed Random Vectors

Consider an $n$-dimensional normal random vector $\mathbf X:= (X_1, \dots, X_n)^T$ with mean $\mathbf 0$ and covariance matrix $\mathbf \Sigma$. Now define a new random vector $\mathbf Y:= (a_1X_1, \...
0
votes
1answer
54 views

Finding expected number of trials until we get head given density function?

Suppose we flip a coin with a random probability of Heads $P$ that has density $f(p) = 6p(1−p),\; p \in [0, 1]$. If we keep on flipping this coin until we get a single Heads, what is the expected ...
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2answers
50 views

Get the distribution of $X|Y=y$ given this joint probability density function

Given the joint probability density function $f(x,y) = \lambda^2 \exp(-\lambda y)$ with $0 < x < y.$ How do I get the distribution of $X|Y=y$ ? Thanks in advance!
3
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1answer
154 views

Find a probability density

I am going through a paper trying to understand all the single steps, but I got stuck. I need to calculate $$p(x+\delta t) \mid x(t), t)= \int p(x(t+\delta t) \mid \mu , x(t), t)p(\mu\mid x(t), t) d\...
1
vote
1answer
77 views

Mixture of Discrete Binomial Distributions

Let $B\left(p,N\right)$ be a Binomial distribution with parameters $p$ and $N$. We define a Mixture of Discrete Binomial Distributions by $\left\{ \left(B\left(p_{i},N\right),\alpha_{i}\right)\right\} ...
3
votes
1answer
40 views

Cancellation law of equal in distribution

I came across this gem while discussing with my friends, If $X$ and $Y$ are two real valued random variables (not necessarily independent) that satisfy $$X =^d X+Y$$ (where $=^d$ means equal in ...
2
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0answers
97 views

Of strings and substrings: A problem of probability

Problem Let $\Sigma=\{a, b\}$. Let $\Sigma^*$ denote the Kleene star of $\Sigma$: \begin{equation*} \Sigma^* = \{\varepsilon, a, b, aa, ab, ba, bb, aaa, aab, \ldots\} \end{equation*} where $\...
2
votes
1answer
33 views

Degree of Polynomial in Centered Moments of Gamma$(n,1)$

I'm interested in the degree of the polynomial in $n$ of the expression for the $k$-th central moment $$ E((X_n - n)^k) $$ where $X_n$ is a Gamma$(n,1)$ random variable, that is, the sum of $n$ ...
0
votes
1answer
91 views

Find the CDF of a function of two random variables

The joint probability density function of two continuous random variables $X$ and $Y$ is: $$f(x,y) = \begin{cases} 6x,& 0\leqslant x\leqslant y,\ 0\leqslant y\leqslant 1\\ 0,& \text{ ...
3
votes
1answer
119 views

Warren's proof for Benford's Law

Warren has a little proof of Benford's law in Hacker's Delight. To quote: Let $f(x)$ for $1 \leq x < 10$ be the probability density function for the leading digits of the set of numbers with ...
2
votes
1answer
63 views

$p$-stable Random Variables for $p>2$?

I will preface this by saying I am certainly no expert in Probability theory. My actual problem is an interpolation one, in which I am considering interpolation of bandlimited functions with shifts ...
1
vote
2answers
39 views

Weighted sum of identical distributed random variables

Suppose $X_1$, $X_2$, $\ldots$ ,$X_N$ are identically distributed (not necessarily independent). Then, given $a_1+a_2+\ldots+a_N=1$, and let $S=a_1 X_1 + a_2 X_2 + \ldots + a_N X_N$. Does $S$ follow ...
2
votes
2answers
114 views

Convergence in distribution for $\frac{Y}{\sqrt{\lambda}}$

Given a sequence of independent r.v's $\{X_n\}_{n\geq 1}$ such that $P(X_n=x)=\frac{1}{2}$ if $x=-1$ and/or $x=1$ Let $N\in Po(\lambda)$ be independent of $\{X_n\}_{n\geq 1}$ and we set that $Y=X_1+...