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

learn more… | top users | synonyms

0
votes
0answers
8 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
votes
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
votes
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?
0
votes
0answers
12 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 ...
-1
votes
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 $r>0$ ?
0
votes
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 ...
1
vote
0answers
22 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) ...
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
votes
0answers
10 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 ...
0
votes
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 & ...
0
votes
0answers
18 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 ...
-1
votes
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
votes
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 ...
-1
votes
1answer
22 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 ...
2
votes
1answer
45 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
22 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
votes
1answer
26 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 ...
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
votes
0answers
8 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 $U_{(1)}=\min U_i$. ...
-2
votes
0answers
59 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
votes
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 ...
1
vote
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 ...
0
votes
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
votes
2answers
42 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
vote
0answers
28 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. ...
1
vote
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
votes
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
votes
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 ...
0
votes
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} * ...
1
vote
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 ...
1
vote
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
votes
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 ...
-6
votes
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
votes
1answer
42 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
votes
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 ...
-2
votes
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
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) ...
0
votes
0answers
17 views

Find mean from geometric PGF

I know that PGF of type 0 geometric random variable is G = p/(1-zs) Now, if I want to find mean, E[X] = d/dz (G) @z=1 = ps/(1-s*z)^2 but according to wikipedia, mean = (1-p)/p and it does not have s ...
0
votes
1answer
21 views

Question on finite dimensional distribution of Markov Chain

If $\{ X_{n} \}$ is a Markov Chain and $X_{o} \sim \pi$ (where $\pi$ is the stationary measure), it follows that the MC is identically distributed. I have a question about the finite dimensional ...
0
votes
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 ...
0
votes
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
votes
1answer
20 views

Distribution Function - Finding $P(X < 3)$ for a given function

this is the same as this: Distribution Function Of a Random Variable X - Question but that question isnt as clear as I was hoping it was The distribution function of the random variable X is given: ...
4
votes
1answer
33 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
votes
0answers
16 views

Support for a linear combination or transformation of random variables

Let $X, Y \sim iid U(0,1)$ and $c_1, c_2 \in \mathbb{R}$. In the linear combination $Z = c_1X+c_2Y$, we know that the probability density function of $Z$ depends on the relationships of $c_1$ and ...
0
votes
0answers
8 views

Is there a analytical formula for Super- and Sub-Poissonian distributions?

I'm currently wrtiting my Bachelors thesis on photon statistics. The way different sources of light can be classified is by Poissonian (coherent light), Super-Poissonian (thermal light) and ...
-1
votes
0answers
29 views

To determine probability distribution for large $N$ with mean at m [on hold]

To show that the following expression turns to Gaussian for large value of $N$ $$\binom{N}{S}\binom{X-1}{a}\binom{Y-1}{a-1}$$ where X+Y+S=N. To show it shows normal distribution with mean at 'm' and ...
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
votes
0answers
16 views

When Independence $\Rightarrow$ Independence of higher moments (Prob/ Stats)

suppose {$X_n$} is iid. Then, is $X_i$ independent of $X_j^3$ for j≠i? If so, why? Secondly, is $X_i^2$ independent of $X_j^2$ for j≠i? Intuition: yes no If there's a difference, why? (note: ...
0
votes
1answer
24 views

Maximum of independent Erlang random variables?

Suppose $Y=\max\{X_1, X_2,\dots,X_N\}$ where all $X_i$ are independent and follows Erlang distribution. I know that extreme value theory deals with maximum of random variables. Can anybody tell me, ...