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

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2answers
46 views

Central Limit Theorem for exponential distribution

Suppose that $X_1$ ..... $X_n$ are a random sample from a population having an exponential distribution with rate parameter $\lambda$. Use the Central Limit Theorem to show that, for large n, ...
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2answers
35 views

Let (X1,…,Xr) ∼ Mult(n,r,p1,…,pr). Determine whether X1 and X2 are independent.

So I'm thinking that I need to have P(X1=k)P(X2=m) = P(X1=k n X2=m), but I'm stuck on where to go from here.
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0answers
9 views

Perfect secrecy not depending on probability distribution of message space

Say we have: $D$ - probability distribution on the messages space $M$. $M_1$ - a random variable of the messages under $D$. $C_1$ - a random variable of the encryption under $D$ (and some ...
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0answers
25 views

To obtain the probability density function of a complicated function of six independent random varibles

Here is the super complicated function: $$Z = \frac{{\left( {CE + DF + 1} \right)\left( {CA + DB + 1} \right)}}{{ABCD + \left( {CA + DB + 1} \right)}}$$ where $A,B,C,D,E,F$ are independent random ...
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1answer
31 views

Probability of making a claim

For an individual, the probability of having an accident in a period of 24 hrs (and therefore the probability of making a claim) is 0.00037. Claims on successive days are independent, and a person can ...
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2answers
26 views

Binomial probability expected value

"A airline charges $560 for a ticket to a popular destination. There are 80 seats on the plane, and the probability that any particular customer will miss their flight is 20%. The airline overbooks ...
0
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1answer
17 views

When to assume exponential distribution?

I have a system which can service requests at constant rate (x requests / s for example). The exponential distribution has this property rate: the average number ...
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0answers
18 views

Find a Borel function

I have trouble understanding what is a random variable. The problem arose when I wondered: Let $X$ and $Y$ be independent and equally distributed random variables. Find a Borel function $B$ such that ...
3
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2answers
31 views

Distributions with finite number of moments

Is it easy to provide an example of a distribution that has, say, finite moments of order one and two, but such that $\mathbb{E}[X^k]=\infty$ or does not exist for all $k>2$ (where $k$ is not ...
2
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1answer
47 views

Find the density of the random variable with characteristic function $\varphi(t)=(1-|t|)^+$.

Suppose a $X$ is a random variable, I am asked to find the density of the random variable with characteristic function $\varphi(t)=(1-|t|)^+$. I am trying to use the inversion formula for the ...
0
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1answer
32 views

Mutual or pairwise independence needed? Variance of a sum.

This is a simple question: Do we need mutual independence or only pairwise independence in order to state that $$\mathrm{Var}\left[\sum_{i=1}^n X_i\right] = \sum_{i=1}^n ...
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0answers
21 views

Order statistics of mixed (iid as well as non iid) random variables

Does anyone know if there are results (PDF or the CDF) on the order statistics (at least minimum or maximum) of $n$ random variables in which a few of them are i.i.d. and the rest of them are ...
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0answers
16 views

Entropy of Sum vs Difference of Random Variable

I am looking for a proof of the following fact Let X and X' be i.i.d on {0,1,2}(not necessarily uniform). Prove that $H(X + X' mod\;3) \leq H(X - X' mod\;3)$ where $H()$ is the standard Shannon ...
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0answers
15 views

Example of caculation the cummulative probability

I have a question about probability that I am confusing. I have k bit symbols. Now, I want to calculate the successful decoding probability for k bits. It can defined by this equation ...
0
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1answer
12 views

Is the linear transform of joint gaussian necessary gaussian? See this case!

Suppose we map the low dimensional Gaussian distribution into higher dimension using linear transform. Say, $X \in R^p$ is joint Gaussian, and for $n > p$, $Y = A_{n \times p}X$. Is $Y$ joint ...
1
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2answers
29 views

show that $P(X=n) = \frac{4}{n(n+1)(n+2)}$ is pmf, that is show that$\sum^{\infty}_{n=1} P(X = n) = 1$

show that $P(X=n) = \frac{4}{n(n+1)(n+2)}$ is a pmf, that is show that$\sum^{\infty}_{n=1} P(X = n) = 1$ My solution $\sum^{\infty}_{n=1} P(X = n)$ $ =\sum^{\infty}_{n=1} \frac{4}{n(n+1)(n+2)}$ $ ...
0
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1answer
19 views

What is the probability distribution described by this problem?

Suppose I have a probability of winning a coin toss 90% of the time; it's an extremely unfair coin :). I want to simulate the number of wins after a specified number of coin tosses. I want to write ...
0
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1answer
28 views

Which Statistical Model to describe data?

Hello, I would really appreciate if someone could help me out with this question. I believe this is a Poisson Distribution because many of the properties are satisfied with this data. There is no ...
1
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1answer
21 views

Find the conditional density function for the following experiment

"A dart is thrown at a circular target of radius $10$ inches, given that it falls in the upper half of the target." I know that a conditional density function is given by the formula $$f(x|E) = ...
1
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1answer
23 views

two soft questions about stochastical ordering

I have two questions and I will be very happy to hear your comments: a-) For two random variables $X$ and $Y$ let $X$ dominate $Y$, i.e. $X>_{ST}Y$. let $f$ be a positive function. Is it true ...
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0answers
33 views

How to show that a closed ball doesn't allow stochastical ordering

Given a closed ball $$\cal{F}=\{g:D(g,f)\leq\epsilon\}$$ where $f$ and $g$ are some density functions and $D$ some distance say relative entropy: $$D(g,f)=\int ...
0
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1answer
28 views

probability: Bernoulli random vector where $X_i$'s all have the same mean and variance

If $\mathbf{X}$ is a Bernoulli random vector where $X_i$'s all have the same mean and variance, is it possible to tell if the $cov(X_i, X_j) = 0$ for $i\neq j$; that is, $X_i$'s are de-correlated? We ...
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0answers
21 views

M/M/1 queue, probability of time spending in queue..

Let $W$ be the time $nth$ customer spends in the queue when $n$ go to $\infty$. How do we write down the formula for $P[W \le t | N = k]$ ? where $N$ is the number of customer in the system.
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1answer
21 views

How can I find the density function of Z?

I am trying to find the density function for Z, this is what I am doing but I am not getting an appropiate function, I don´t know if there is something wrong with limits of the intregral. Or if this ...
0
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1answer
15 views

Probability question involving Exponential and Poisson distributions

The number of insects hitting a windscreen of a car travelling from Johannesburg to Nelspruit follows a Poisson distribution. The variance of the waiting time between insect strikes is 36 ...
2
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3answers
128 views

How to compute the powers of $2\times2$ Markov matrices

Consider a Markov chain $(X_n)$ with state space $E=\{1,2\}$. If given a transition matrix $$P=\pmatrix{1-a&a\\b&1-b}\;,$$ with $0<a,b<1$. How to find out the $n$-th power to the ...
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1answer
34 views

Challenging Question: for Expected Value of a particular probability density function

I've been stuck on this for a while and it's been driving me crazy. Any help would be greatly appreciated. I am trying to find the Expected Value of the following Probability Density Functions (where ...
0
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0answers
40 views

symmetric and anti symmeric distribution - sqrt function on it

I've got question for homework and I'm not sure about it, I appreciate your help. 1. assuming distribution is anti symmetric, if we apply the function sqrt on it, will we get symmetric distribution ...
0
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1answer
14 views

Poisson Distribution with a maximum data value

Here is the question: A shopkeeper hires vacuum cleaners to the public at £5 per day. The mean daily demand is 2.6. If only 3 vacuum cleaners are available for hire calculate the mean of the daily ...
2
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0answers
59 views

Let $X_n \sim U[0,1]$. Let $A_n$ count the number of local maxima of the sequence unto $n$. Prove a suitable central limit theorem for $A_n$.

Let $X_n $ be uniformly distributed on $[0,1]$. We say $X_k$ is a local maximum if $X_k> X_{k\pm 1}$. Let $A_n$ count the number of local maxima of the sequence unto and including $n$. Find $a_n, ...
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0answers
10 views

Calculating the probability mass function of the sum of two independent, non-similar, geometric random variables using convolution

Given two independent geometric random variables (where $p_1,p_2 \in [0,1]$): $$\mathbf{X_1} \sim \text{geometric}(p_1)$$ $$\mathbf{X_2} \sim \text{geometric}(p_2)$$ I want to find the probability ...
0
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0answers
11 views

Finding the conditional density of some Chi-Squared distribution.

So let's say: $X_1,X_2,...$ are standard normal variables, and I form: $T_n = X_1^2 + ... + X_n^2$ $T_n$ would have $\chi^2$ density. Now I fix $T_n = t$ and would like to find the conditional ...
2
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2answers
72 views

Evaluating a double integral from zero to infinity

How do I evaluate this integral? I don't understand at which point the limit notation should set in? And my method yields $0$ in the end. The integral is: $$ \int_0^{\infty} \int_0^{\infty} ...
-1
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0answers
13 views

Calculate the marginal p.d.f of Y, f2(y), when X~U(0,2) and the conditional distribution of Y, given X = x is U(0,x^2)

Let X have a uniform distribution U(0,2), and let the conditional distribution of Y, given that X = x, be U(0,x^2). a) Determine f(x,y), the joint p.d.f. of X and Y: f1(x)=1 0<=x<=2 ...
0
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1answer
15 views

Easy Question about Computing the probability of properties of random subsets

I want to solve the following exercise: Suppose that two sets $X$ and $Y$ are chosen independently and uniformly at random from all the $2^n$ subsets of $\{1, \dotsc, n\}$. Determine $P[X \subseteq ...
1
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1answer
23 views

Probability of an event

Problem. Suppose we have $n + 1$ random variable $\xi_0, \xi_1, \dots, \xi_n$ and they are independent and all standard normal distributed. Find probability that $\xi_0$ greater than $\max\{\xi_1, ...
3
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0answers
48 views

Question about application of Erdős-Kac theorem

My question is whether (*) below can be shown using the Erdős-Kac theorem? I don't think the distinction between $\Omega$ and $\omega$ is important here. For lack of better notation let ...
3
votes
1answer
39 views

How to calculate the characteristic function of compound Poisson random variable?

Let $\phi_X(t)$ be the characteristic function of $X$. Let $N$ be a Poisson random varivale with mean $1$ and $(X_i)_{\in\mathbb{N}}$ be i.i.d. copies of $X$. Then how to derive the charactersitics ...
1
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1answer
16 views

Calculating probability using normal tables

I've had a crack at this question however I don't seem to be getting the correct answer and I can't figure out why. I've been given a table of the 'Normal Distribution Function' where the left tail is ...
1
vote
1answer
23 views

continuous probability: signal to noise ratio $\mu^2/\sigma^2$

$\DeclareMathOperator{\var}{var}\DeclareMathOperator{\cov}{cov}$ The signal-to-noise ratio (SNR) of a random variable quantifies the accuracy of a measurement of a physical quantity. It is defined ...
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0answers
20 views

Probability Transformation

Let the continuous random variables $X$ and $Y$ have the joint probability density function given by $f(x)=3/2x$ for $0<x<2$, $0<y<1$, $x<2y$. Find the probability density function ...
3
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1answer
23 views

Bernoulli event expansion in 0 to 3 occurrences cancel to first order

I am working through Hamming's The Art of Probability and am having trouble with a problem in the Bernouilli Trials section. The wording is the following Expand the binomials in the probabilities ...
0
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1answer
48 views

what is the probablity that all the lengths are smaller than $a$?

I met an interesting but challenging problem in my homework: Suppose $n-1$ independent points $x_1$, $x_2$, ..., $x_{n-1}$ are uniformly distributed on unit interval [0,1]. These $n-1$ points ...
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1answer
60 views

In a game you receive three cards, ω, from a well-shuffled deck.. [closed]

In a game you receive three cards, ω, from a well-shuffled deck. You then receive \$30 per face card contained in the hand. That is if the hand contains 1 face card you get \$30, 2 you get \$80, 3 ...
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0answers
12 views

Confidence interval issue

n (sample size) = 96 Average value x¯ = 4.1 σ = 4.5 I have to calculate the confidence interval's lower endpoint a and upper endpoint b for expected value 99,5%. (Normal distributed). So far I ...
1
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2answers
23 views

Conditional Coin Probability:Will The Decision Change

A decision making problem will be resolved by tossing $2n + 1$ coins. If Head comes in majority one option will be taken, for majority of tails it’ll be the other one. Initially all the coins were ...
1
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1answer
37 views

Probabability of Joint Distribution

Let the continuous random variables $X$ and $Y$ have the joint probability density function given by $f(x) = 3/2x$ for $0<x<2$, $0<y<1$, $x<2y$. Find Pr $(x<1.5|y>0.5)$. This was ...
4
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0answers
87 views

How to model this easy problem as sum of indicator random variables in order to apply Chernoff bound

Do you have an idea how I could model the following process somehow as a sum of independent indicator random variables? I have given a grid of size $n \times n$ for $n \rightarrow \infty$. Now I ...
0
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2answers
22 views

Inequality in proof that $ X_n \overset{P} \to X \Rightarrow X_n \overset{d} \to X $

I am looking at the proof of convergence in probability implying convergence in distribution. The proof begins by stating that if $X_n \leq x$ then either $ X \leq x + \epsilon $ or $ |X_n - X| > ...
1
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1answer
25 views

Chi square distribution

K is chi-square distributed with 33 as degree of freedom. P(x ≤ K ≤ y) = 0,73 and the probability of the above and below [x,y] is equal. I'm supposed to define x and y. I tried solving it like this: ...