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

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1answer
17 views

Finding the variance of the time series defined as $x_t=\phi x_{t-1}+w_t$, for $t=2,3,4,…$.

Let $w_t$ be white noise with variance $\sigma_w^2$ and let $|\phi|<1$ be a constant. Consider the process $x_t=w_1$ and $x_t=\phi x_{t-1}+w_t$ for $t=2,3,...$. I need to find the variance. I ...
-2
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1answer
68 views

Discrete distribution problem in medical application

im having trouble with question 2 on one of my math papers. I would greatly appreciate it if someone could help me out here, preferably give me worked out solutions for this question. Thank you for ...
0
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0answers
10 views

Normal distribution and covariance matrix

The question is as follows: Consider a normal distribution with mean $\mu = 0$ and covariance matrix $\sum$. Let $v$ be an eigen vector of $\sum$ with eigen value $\lambda$. What does it mean ...
0
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0answers
13 views

Find probability distribution knowing the conditional probability distribution

I have been working on some physics problem which I "translated" to the following mathematical problem for which I need help to solve: Suppose we have the random variable $\alpha$ distributed ...
-4
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1answer
10 views

Maximizing Varience of Independent Random Variables [on hold]

Suppose X and Y are independent mean 0 random variables, with positive variances a and b, respectively. Find the value of c that minimizes the variance of cX+(1-c)Y?
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0answers
21 views

Inequality with poisson distribution [on hold]

Let $r>1$ and $X \sim Poiss(\lambda)$. Prove that $$ \mathbb{E} X^r \le r^r + (e \cdot \lambda)^r $$ Does this inequality hold for $r>0$ ?
2
votes
1answer
483 views

Affine transform of multivariate gaussian

If $X_1, \ldots, X_n$ are iid $N(0,1)$ or in other words $\mathbf{X}=(X_1, \ldots, X_n)$ is distributed $N(\mathbf{0}, \mathbf{I})$, then $A\mathbf{X}+\mu$ is distributed $N(\mu, AA^t)$. Showing that ...
0
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1answer
31 views

Chebyshev's inequality for 1 standard deviation results in 0?

In applying Chebyshev's inequality to a probability distribution, the following is the given equation: $$p(\mu - c*\sigma \le X \le \mu + c*\sigma) \ge 1 - \frac{1}{c^2}$$ This indicates for any ...
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0answers
25 views

Random variable: $X\sim Normal(m, {\sigma}^2)$, find the characteristic function of $X^2$

Is it possible, knowing that $X$ is a random variable with normal distribution( with parameters $(m, {\sigma}^2)$), to find the characteristic function of $X^2$ ? What I thought is: Since: $\phi(X) ...
0
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0answers
7 views

Stationary distribution of a stochastic process

I have a discrete time stochastic process $\{X_t : t \in T\}$ with continuous state space. Assume $X_0=0$ and increments $\delta_t$ are exponential with mean $\alpha$ (so its parameter is ...
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1answer
13 views

Find marginal probability distribution of $ X$?

$X$ and $Y$ have a bivariate normal distribution with $\sigma_X= 5\ mL,\ \sigma_Y= 2\ mL, \ \mu_X= 120\ mL, \ \mu_Y= 100\ mL$ and $\rho = 0.6.$ How do I find the marginal probability distribution of ...
0
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1answer
28 views

What is the probability that a multivariate Gaussian random variable is greater than zero?

I am looking for a way to find the probability that $p(x > 0)$, where the vector $x$ has a multivariate Gaussian distribution $$ x = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \sim ...
1
vote
1answer
21 views

Determine the probability density function of…

Let $X$ be a random variable with normal distribution with parameters: $$m = 1$$ and $$\sigma = 2$$ How can the probability density function of $$Z = -\frac{\ln |X|}{3}$$ be determined?
0
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0answers
12 views

moment generating function with Taylor series simplification

Denotes $a(0,r_1,r_2)$ as the annulus with radii $r_1<r_2$ centered at the origin $0$ Consider two bands $a(0,s,t)$ and $a(0,u,\sim)$ for $1\leq s\leq t\leq u$ Suppose a variable (call it an ...
1
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1answer
45 views

Is the following probability distribution stationary/constant

For a conservative system, we know that angular momentum, $l$, and total energy, $E$, are constant, i.e. $\dot{l}=\frac{dl}{dt} = 0$ and $\dot{E}=\frac{dE}{dt} = 0$, where $t$ indicates time. Let ...
0
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1answer
21 views

find distribution of $\max(x^2,x)$ and $\min (x,1)$

I have the following question. Find distribution of $Y=\max(X^2,X)$ and $Z=\min(X,1)$. My distribution function is $$ F_X(x)=\left\{\begin{array}{ll} 0 & \mathrm{if}\; x <0\\ 0.5x & ...
1
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1answer
30 views

Distribution of an angle between a random and fixed unit-length $n$-vectors

Suppose I have a random unit-length $n$-element vector $\mathbf{x}$ that is uniformly distributed on an $n$-dimensional sphere, and let vector $\mathbf{a}$ be some other unit-length $n$-element vector ...
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1answer
27 views

Lottery Distributions Question [on hold]

In a certain lottery, $7$ balls are drawn at random (without replacement) from $n$ balls numbered $1$ through $n$. Let $P$ be the probability that no pair of consecutive numbers is drawn. Let $Q$ be ...
0
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1answer
19 views

Probability density function for a PDE with random inputs

I am looking for a general method or alternatively few textbook examples of deriving a probability density function for a solution of partial differential equation with random inputs in the equation ...
0
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0answers
61 views
+50

Stochastic dominance of Binomial and Poission

In order to investigate the size of the cluster of a given vetex in a random graph I need to use a fact about stochastic dominance that I don't know how to prove. Namely, I am looking for a proof of ...
0
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0answers
19 views

Geometric Distribution

The police have stated that 20% of the items sold by pawn shops in the city have been stolen. Ralph has just purchased 4 items from one of the city’s pawn shops. Assuming the official is correct, and ...
2
votes
1answer
46 views

Random variable with infinite expectation

I was trying to find $Y$,a random variable (non-negative, may be $E(|Y|)=\infty$), such that $$\sum_{n=0}^{\infty} E\Bigl(\frac{|Y|}{n^2 +|Y|}\Bigr)=\infty$$ I tried with Cauchy distribution but could ...
1
vote
1answer
12 views

Average Waiting Time for mixed distribution function

Mixed Distribution Function $$ F(t) = \begin{cases} \hfill 0 \hfill & t < 0 \\ \hfill p+(1-p)(1-e^{-yt}) & t \geq 0 \end{cases} $$ How can i find the average waiting time of an ...
2
votes
1answer
23 views

What is the pdf and cdf of $aX^2+bX$?

If $X$ is normally distributed, $X \sim N(0, \sigma) $, what distribution is $aX^2+bX$? Is there any way to express the cdf and pdf? Thanks.
0
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1answer
25 views

Poisson arrival times joint distribution

The arrival times of the first and second event are $S_1$ and $S_2$, and the number of arrivals follow a poisson process. How would I compute the Joint PDF of $S_1$ and $S_2$? I have found the PDF of ...
0
votes
0answers
13 views

Support of Distribution Function

Suppose I have a distribution function $$C(u,v)$$ with domain $I^2$. Let us define the support of this function as the complement of the union of all open subsets of $I^2$ with C-measure zero. Based ...
0
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0answers
9 views

Distribution of $\sum n_i(U_i-U_{(1)})$

Let $U_i$ be independent random variables with pdf $f_i(x)$ ($i=1,\ldots,k$) where $$f_i(x)=\frac{n_i}{\sigma}\exp(-\frac{xn_i}{\sigma}), x>0$$ Let $n=\sum n_i$ and $U_{(1)}=\min U_i$. ...
3
votes
2answers
416 views

Poisson Distribution when only given using mean

I'm doing the following homework problem and am unsure of whether or not my answers are correct. This is my first time working with Poisson distribution and I want to make sure I am doing it ...
0
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2answers
47 views

To use or not Bernoulli trials

I was asked to model the following experiment: Consider the n-th toss of a fair coin, and the event $E$ = '$k$-th toss results in heads'. I find easier to model the experiment using n random ...
1
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0answers
18 views

Continuous Distribution [on hold]

Suppose we observe the value of a random variable $X$ with pdf $f(x)=2x$, $0<x<1$. The value divides the interval $(0,1)$ into two subintervals. Let $Z= \min[X, 1-X]$ be the length of the ...
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0answers
9 views

Maximum likelihood estimates for exponential distribution [on hold]

If the random variable 'x' has the following PDF $f(x)=\beta e^{-\beta (x-\alpha)}, x\geq \alpha, \beta >0.$ What will be the maximum likelihood estimates for both parameters $\alpha$ and ...
0
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1answer
20 views

Frequency Distribution and Throughput

I am conducting an experiment on a couple of computer systems but the results I have don't make sense to me. I made each system perform 1000 operations: System A performs operations at a rate of ...
1
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1answer
383 views

Find the MOM estimate and the MLE of the Pareto distribution.

The Pareto distribution has been used in economics as a model for a density function with a slowly decaying tail: Assume that $X_0$ > 0 is given and that $X_1, X_2, ..., X_n$ is an i.i.d. sample. ...
8
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2answers
7k views

Proof of $\frac{(n-1)S^2}{\sigma^2} \backsim \chi^2_{n-1}$

It's a standard result that given $X_1,\cdots ,X_n $ random sample from $N(\mu,\sigma^2)$, the random variable $$\frac{(n-1)S^2}{\sigma^2}$$ has a chi-square distribution with $(n-1)$ degrees of ...
0
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2answers
44 views

Random Variables and Probability Distributions

Little Help here Q-For a laboratory assignment, if the equipment is working, the density function of observed outcome X is f(x) = 2(1-x), 0 < x< 1 otherwise 0 ...
1
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0answers
29 views

Flip a coin, then repeat an experiment n times. Show exchangeable but not independent

We flip a fair coin. If it is heads then we roll a die n times, if it is tails we sample a number n times from the set {1, 2, 3, 4} with replacement. We denote the resulting n numbers by X1, ..., Xn. ...
2
votes
2answers
26 views

Find asymptotic variance MLE heavy tailed distribution

$$\mathbf{X} = \{X_1,X_2,\dots,X_n\}$$ sequence of i.i.d. RV's. Let the distribution of the RV's be defined by $$f(x|\theta)=\frac{\theta}{x^{\theta+1}}, \quad x>1, \quad \theta>1$$ I am ...
1
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0answers
22 views

The convergence of probability for $X_nY_n$ and $X_n/Y_n$

Suppose that $X_n, Yn$ ($Y_n\neq 0$ a.s) converge to $X,Y$,respectively, in probability. I need to show 1) $X_nY_n \rightarrow XY$ in probability. 2) $X_n/Y_n \rightarrow X/Y$ in probability. My ...
0
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1answer
18 views

Equivalent of random variable sequences in distribution?

Suppose that $X_n, Y_n$ are sequences of random variable on probability space $\Omega$. If $Xn,Yn$ converges to $X$ ( some random variable ) in distribution, then is $X_n=Y_n$ almost everywhere (a.s)? ...
0
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0answers
28 views

To determine asymptotic value of funtion for large N

To show that the function follows normal Gaussian for large value of N (s.t. m is much less than N ) with mean at 'm'. $f(m,N)=\sum_{a=1}^{\lfloor N/2 \rfloor} \binom{N}{S} * ...
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0answers
16 views

Expectation of inverse of sum of random variables, exponential distribution

I have a question similar to this one: Expectation of inverse of sum of random variables only my variables have exponential distribution. So $X_1, X_2, ...$ are $iid$ with exponential distribution. ...
2
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1answer
46 views

Expected Payoffs

In simple setting consider revenue is dependent on variable $w$ which is uniformly distributed $[0,1]$. The revenue function is $wd$, where $d$ is development program. How to I get the expected ...
4
votes
2answers
73 views

For X,Y random variables, with pdfs that are symmetric around 0, does $V(X)\geq V(Y) \Rightarrow E(|X|)\geq E(|Y|)$?

I need to show the following thing. Consider two continuous random variables $X,Y$ which take values in $[-1,1]$ and are have pdf's that are symmetric around zero. How can I show that $V(X)\geq V(Y) ...
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0answers
30 views

The probability that Z is between 0 and -1.61 [on hold]

What is the probability that Z is between 0 and -1.61? Would be good if you could show working. Thanks.
-1
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1answer
44 views

There are 20 red marbles, 10 blue marbles, and 5 white marbles in a jar.

There are 20 red marbles, 10 blue marbles, and 5 white marbles in a jar. Select a marble without looking, note the color, and then replace the marble in the jar. We’re interested in the number of ...
0
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0answers
22 views

Relationship between distributions of correlations $\rho(X^1,Y^1)$ and $\rho(X^2,Y^2)$ if $X^2=WX^1$, $Y^2=WY^1$ and $W$ is a known stochastic matrix?

I have been stacked for a while with the following problem: Consider two samples of iid observations $X^1=\{X_1^1,\dots,X_n^1\}$ and $Y_1=\{Y_1^1,\dots,Y_n^1\}$ where $X_i^1 \sim \mathcal{N}(0,1)$ and ...
0
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1answer
20 views

What's the interpretation of this random variable

Let $(\Omega,\mathscr{F},P)$ be a probability space and $X$ be a random variable that takes values in $\mathbb{N}$. Define $$q(n)\equiv P(X=n)\quad n\in\mathbb{N}$.$$ So $q$ is just the probability ...
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0answers
30 views

Probability distribution function and probability density function [on hold]

In constructing the bridge shown below, an engineer is concerned with forces acting on the end supports caused by a randomly applied concentrated load P, the term ‘randomly applied’ meaning that the ...
4
votes
1answer
34 views

Deriving master equation for discrete process

Consider a group of $N$ professors, $Y$ of whom are wearing white socks and $X = N − Y$ others who are wearing black socks. On each time step, one professor is chosen at random and he has to put a new ...
0
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1answer
12 views

What is the maximum of $n$ points with CDF $F$ and PDF $f$?

I read somewhere that the minimum of $n$ points with CDF $F$ and PDF $f$ is $g(y) = n(1-F(y))^{(n-1)}f(y)$ What would the corresponding maximum value of the points be? Also, how do we derive the ...