Questions on using, finding, or otherwise relating to probability distributions, pdfs, cdfs, or the like.
1
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2answers
75 views
Distribution of sum of independent random variables
Let $X$ and $Y$ be independent random variables taking values in $[0,1]$ where $X$ is uniform. Question is, what distribution on $Y$ will yield a uniform distribution on $[0,2]$ for the sum $Z=X+Y$?
...
1
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2answers
54 views
calculate probabilty, Uniform distribution
this is my first question so excuse my unknowing and mistakes:
I was reading a book and just faced this thing:
(1.4) $=P(X\gt Z/2)(Y-X)$
(1.5) $=P(2X\gt Z)(Y-X)$
(1.6) $=\min\{{2X,1\}}(Y-X)$
...
0
votes
3answers
48 views
if we flip the coin $100$ times, what is$ P(X\leq 10)$?
we have a coin of diameter $d$ and a table of infinite grid of identical squares, each square has side $s$. suppose that $2d = s$. let $X$ denote the total number of times that the coin ends up within ...
0
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1answer
181 views
Find the conditional probability density function.
If I am given X that follows an exponential distribution with mean m and Y that follows a poisson distribution with mean n, how can I use them to find the conditional probability density function of X ...
0
votes
1answer
37 views
Probability density function for the normalised sum of N random variables
I was wondering what the PDF looks like for Z= (1/N)*SUM(z_1+...+ z_n), where each z_i is computationally represented by RAND(). What is the behaviour of the PDF as N -> infinity?
0
votes
1answer
289 views
expected value calculation for squared normal distribution
I need help with the following problem. Suppose $Z=N(0,s)$ i.e. normally distributed random variable with standard deviation $\sqrt{s}$. I need to calculate $E[Z^2]$. My attempt is to do something ...
1
vote
2answers
128 views
Probability density function of a product of uniform random variables
Let $z = xy$ be a product of two uniform random variables, with $x$ having the range $[a, b)$ and $y$ the range $[c, d)$.
What is the probability density function of $z$, and how is it calculated?
1
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1answer
12 views
Distribution of the product of two independent and complex Gaussian Random Variables
What is the distribution of the product of two independent and complex Gaussian Random Variables? Assume they both have zero mean.
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1answer
88 views
Distribution of minimum and sum of two independent exponential random variables
How can I solve this problem?
Is there any formula for this problem
Find the distribution of the random variable $Y$ if
$Y=\min(X_1,X_2)$
$Y=X_1+X_2$
where $X_1$ and $X_2$ are independent ...
1
vote
1answer
409 views
Variance for a product-normal distribution
I have two normally distributed random variables (zero mean), and I am interested in the distribution of their product; a normal product distribution.
It's a strange distribution involving a delta ...
0
votes
1answer
146 views
Use the MFG (Moment Generating Function) technique to determine the joint distribution of (X,Y)
Im given V and W are independent standard normal random variables where $x=\frac{(V+W)}{\sqrt(2)}$ and $y=\frac{(V-W)}{\sqrt(2)}$.
This is what I did:
...
0
votes
0answers
16 views
function of a continuous random variable that is a bernoulli trial?
what is a function of a continuous random variable that is a bernoulli trial?
x= continuous random variable
function(x) = bernoulli trial
examples of continuous random variables: exponential, ...
1
vote
2answers
35 views
Approximating a probability from a sample of size $200$
I'm having some issues with this problem.
The median age of residents of US people is $35.6$ years. If a survey of $200$ residents is taken, approximate the probability that at least $110$ will be ...
0
votes
2answers
42 views
Test of hypothesis
Can you help for solving this question.What ı will use to solve this problem.I try to do something but ı thınk not correct.ı have an exam please help me
1
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1answer
60 views
Discrete Pareto Distribution
What would be the equation to describe a set of 10,000 bugs if you already know they have a pareto distribution.
In other words 2,000 of the bugs would equal 80% of all your problems.
I'm struggling ...
0
votes
0answers
28 views
Exponential Order Statistics Independence
Are the order statistics from the $n$-sample with $X_i\sim \text{Exp}(\lambda)$ (taking, without loss of generality, $\lambda=1$) $\Delta_{(k)}X=X_{(k)}-X_{(k-1)}$ independent?
Can show that for an ...
1
vote
0answers
24 views
Calculate Expectancy of Poisson distribution with Poisson parameter
The problem:
Let X have the Poisson distribution with parameter $\lambda$, where $\lambda >0$.
Calculate E[X!] for every possible value of $\lambda$,
So what I got so far:
$$E[X!] = ...
1
vote
2answers
38 views
Diffusion process. Distribution vs transition probability.
I need confirmation on the following problem: Take a SDE of the form:
\begin{equation}
dX_t=a(X_t,t)dt+b(X_t,t)dW_t
\end{equation}
where all the conditions, such that the solution $X_t$ is defined ...
-1
votes
1answer
28 views
Proof for one of the Cumulative Distribution Function
Hey I tried searching online but I can't find this proof based on the CDF.
$$P(a < X \le b) = F(b) - F(a)$$
Thanks for any help.
0
votes
1answer
34 views
Moment generating function using probability function?
Suppose that $X_1, X_2, ..., X_n$ are independent, where each $X_i$ has probability (mass) function $p_i(x_i)$ given as $p_i(x_i) = \frac{e^{-\lambda} \lambda_i^{x_i}}{x_i!}$ (only the parameter ...
1
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0answers
19 views
Estimate on Galton-Watson process distribution
Let $(Zn)_{n\in \mathbb N_0}$ be a Galton-Watson process, i.e.
$$ Z_{n+1} = \sum_{k=1}^{Z_n}\xi_{n,k},\qquad (\xi_{n,k})_{n\in \mathbb N_0,k \in \mathbb N} \quad \text{i.i.d } \mathbb N_0 \text{ ...
0
votes
2answers
62 views
Conditional Expectation of Exponential Order Statistic $\text{E}(X_{(2)} \mid X_{(1)}=r_1)$
Having already worked out the distributions of $\Delta_{(2)}X=X_{(2)}-X_{(1)}\sim\text{Exp}(\lambda)$ and of $\Delta_{(1)}X=X_{(1)}\sim\text{Exp}(2\lambda)$ where $X_{(i)}$ are the $i$th order ...
0
votes
1answer
36 views
deriving the distribution of Y having information about X
A child who has swallowed a pin of length 4cm is X-rayed. THe pin appears on the X-ray film as shown below, the length of the image being y, an observation on the random variable Y. The pin is at an ...
2
votes
1answer
230 views
Conditional Moment Generating Function With A Twist
Let $X$, $X'$ be identically distributed (not necessarily iid) random variables with compact support, on the same probability space. Define
$G_t(x):=\mathbb{E}[e^{t(X'-X)} | X=x]$
In other words a ...
2
votes
2answers
39 views
Clarification on expected number of coin tosses for specific outcomes.
As seen in this question, André Nicolas provides a solution for 5 heads in a row.
Basically, for any sort of problem that relies on determining this sort of probability, if the chance of each event ...
0
votes
1answer
47 views
Confused between normal and binomial dist.
The fish in a lake have weights that are normally distributed with a mean of 1.3 kg and
a standard deviation of 0.2 kg.
(b) John catches 6 fish. Calculate the probability that at least 4 of the fish ...
3
votes
2answers
41 views
Probability distribution of product of integers
I have a scoring system based on 5 factors with integer values from 1 to 5:
Score = A * B * C * D * E
So the Score can range from 1 to 3125. Each of the factors ...
2
votes
2answers
44 views
Understanding $P(X=n)=0$ when $X$ follows a continuous distribution
I have something that I can't get about continuous distribution. Let's say a variable $X$ follows a continuous distribution on $]0;1[$. Then when I pick a completely random number on $]0;1[$, say I ...
0
votes
1answer
39 views
functions of several random variables
Let $R$ and $X$ be independent non-negative random variables such that
$R^2\sim \chi^2_2$ and $X\sim U(0, 2\pi)$. Fix $a$ belonging to $(0, 2\pi)$. Find the distribution of $R\sin(X+a)$.
1
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1answer
61 views
cdf of sum of a discrete and continuous random variable
Let U be uniformly distributed on the interval (0, 2) and let V be an independent
random variable which has a discrete uniform distribution on {0, 1, . . . , n}. i.e.
P{V = i} =1/n+1 for i = 0, 1, . . ...
3
votes
1answer
80 views
Exponential Distribution Function
If $X\sim \text{Exp}(X)$ then for all positive $a$ and $b$, $P(X>a+b\mid X>a)=P(X>b).$ So given independent random variables $X \sim \text{Exp}(\lambda)$, $Y \sim\text{Exp}(\mu)$ we would ...
1
vote
1answer
49 views
The probability of no cars passing within a certain time interval
Let's say there is an induction loop in a road capable of counting the number of cars passing over it. By keeping a list of moments for when a car passed the detection loop, I am able to determine the ...
0
votes
1answer
65 views
Probability distribution question
Suppose that X is Poisson distributed with mean $\lambda >0$ and Y is geometrically distributed with parameter $p\in(0, 1)$.
Assume that X and Y are mutually independent. How do I show that $P(Y ...
1
vote
1answer
178 views
Exponential distribution
Let $X_1 ~ \sim \text{Exp}(\lambda_1)$ and $X_2 \sim \text{Exp}(\lambda_2)$ be two independent exponentially distributed random variables.
Show that $\mathbb{P}\{\min(X_1, X_2) > t\} = ...
1
vote
1answer
39 views
Verify a distribution that is not exponential family
I understand that if the support of a distribution depends on the parameter $\theta$, it is not exponential family even if its pdf can be written in the form $ f(x | \theta) = h(x)c(\theta) \exp\left( ...
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0answers
28 views
Gap distribution independence proof
I have a question bout the proof of the independence of gap RVs. Given the independent exponentially distributed random variables $\xi_1$, $\xi_2$ ~ $\text{Exp}(\lambda)$, and a corresponding order ...
1
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0answers
33 views
Convergence in distribution and ordered statistics
Let $X_1,X_2,\ldots,X_n$ be i.i.d from some distribution $F$. Denote the ordered statistic of the previous as $X_{(1)}, X_{(2)},\ldots, X_{(n)}$. For $t \in \mathbb R$, define $Z_n = \frac{i}{n}$ if ...
0
votes
1answer
142 views
probability and simulation for baseball games
In order for a baseball team to win the world series,the team must win four of seven games. assume the two teams are equally likely to win. 1) Using a simulation with 25 trials,what is the probaility ...
2
votes
1answer
159 views
What's the expected value in this jackpot winning experiment.
If there exists a fair National Lottery, that someone bets £1, Jackpot increases by £1, and there is p chance that he wins a Jackpot. If a Jackpot is won, it is reset to 0. repeats.
We can easily ...
0
votes
1answer
35 views
HELP Distribution of the Minimum of two random variables
Well Let $Y$ be a random variable that could be discrete or continuous and $M$ a positive constant random variable Find the distribution of $S$$=$$min${Y,M}
My progress so far is :
$p($S ...
2
votes
1answer
48 views
How to find the probability of one die roll being higher than a second die roll?
And the catch is the first die has $x$ sides, while the second die has $y$ sides, where $x\neq y$
I had this figured out several years ago, but apparently have forgotten both the answer and too much ...
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0answers
16 views
How to marginalize variable in SamIam [closed]
I want to marginalize some variables while calculating marginal distribution in SamIam, but I cannot find that option.
How can I marginalize out variable?
Thanks!
1
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2answers
25 views
Obtain the cumulative distribution function of $X_1+X_2$
Suppose $X_1$ is a standard normal random variable. Define
$$X_2=\begin{cases} -X_1, &\text{if} \,\, |X_1|<1 \\ \,\,\,\,X_1, & \text{otherwise}\end{cases}$$ Obtain the cumulative ...
1
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0answers
28 views
Upper bound on truncation error of a fourier series approximation of a pdf?
Given a probability density function, $f\left(x\right)$, of a continuous random variable, $X$, and given an $N$-th order fourier series approximation:
$$f_N\left(x\right)=\sum_{n=-N}^{N}c_n e^{inx}$$
...
6
votes
1answer
42 views
How can a $\sigma$-algebra be “treated” or computed? Example
My question is: I have a random variable $X:\Omega \rightarrow \mathbb{R}$, the $\sigma$-algebra generated by $X$ is: $\sigma(X) := \{X^{-1}(B), B\in \mathcal{B}(\mathbb{R})\}$.
But, imagine now that ...
0
votes
0answers
93 views
The mean and variance of the random variable with Rician distribution
What is the mean and variance of $Z_0=\frac{\sum\nolimits_{i = 1}^{{n}}R_iZ_i}{\sum\nolimits_{i = 1}^{{n}}R_i}$, where $Z_i$ is a constant and $R_{i} =\sqrt{X_i^2+Y_i^2} $ ${(i=1,\ldots ,n)}$? $X_i$ ...
2
votes
1answer
46 views
Using empirical density function as an estimator of a given probability density
We know empirical distribution function is defined as $F_n(x)=\frac{1}{n}\sum\limits_{i=1}^nI(X_i \leq x)$. Then define empirical density function as $ f_n(x) = \frac{F_n(x+b_n)-F_n(x-b_n)}{2b_n} $ .
...
1
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1answer
26 views
Let $X_1,\ldots,X_n$ be IID from EXP($\lambda=1/ \theta$) so $E[X_1\mid\theta]=\theta$. Use MGF technique to find distribution of $\sum_{i=1}^{n} X_i$
I ended up getting $\left(\dfrac{\lambda}{\lambda-t}\right)^n \sim \operatorname{Gamma}(n, \lambda)$
Is this right, I haven't done the process in a while so if this is wrong I would like to see the ...
2
votes
1answer
57 views
Confusion about Banach Matchbox problem
While trying to solve Banach matchbox problem, I am getting a wrong answer. I dont understand what mistake I made. Please help me understand.
The problem statement is presented below (Source:Here)
...
0
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0answers
23 views
Variability in estimations over a non-ergodic/non-regular Markov process
Imagine we have a non-ergodic/non-regular Markov Process with with $n$ states.
Among these $n$ states, there are $k$ absorbing states.
For each of the $n-k$ non-absorbing states, it is not possible ...
