Tagged Questions
0
votes
0answers
23 views
Cumulative distribution function of a function of random variable
Random variable $G$ has pdf $f(g)=\frac{2}{3}\cdot e(-2/3g)$ for $g>0$ and $f(g)=0$ otherwise. Now, $L=7$ if $G<5$ and $L=3G$ if G>=5. How to find cumulative distribution function of $L$
1
vote
1answer
33 views
Constructing Distribution By Coin Flipping
I am interested in any example of construction distribution by coin flipping.
Actually I want to show the process of construction a random variable $X$ with distribution $(p_1,...,p_n)$ by coin ...
0
votes
1answer
55 views
Computing PDF of Products of Two Random Variables
I've been stuck on this problem for some days. I'm hoping someone would help by chipping in a few comments. I have two i.i.d. r.v.:
$$
f_X(x)=\frac{\left(1-e^{-\frac{x}{\alpha }}\right)^{\tilde{r}-1} ...
1
vote
0answers
26 views
P.d.f of a discrete fourier transform of binary variables
Let $\{a_n\}$ be a set of $N$ "binary" random variables uniformly distribuited in $\{-1,1\}$.
The discrete fourier transform is defined
$b_k=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1} a_n e^{-2 \pi i k n ...
-4
votes
0answers
23 views
Symmetric random variable
If $X$ and $Y$ are two independent random variables where $Y$ is symmetric about $0$.Let $U=X+Y$ and $V=X-Y$ then $U$ and $V$ have the same distribution.
0
votes
2answers
55 views
Probability distribution of a sum of uniform random variables
Given
$$X = \sum_i^n x_i$$
,where $x_i \in (a_i,b_i)$ are independent uniform random variables, how does one find the probability distribution of $X$.
5
votes
0answers
64 views
What is the distribution of $\sqrt{X^2+Y^2}$ when $X$ and $Y$ are Gaussian but correlated?
If $Z = \sqrt{X^2+Y^2}$, and $X$ and $Y$ are zero-mean i.i.d. normally-distributed random variables, then $Z$ is Rayleigh distributed.
What is the distribution of $Z$ if $X$ and $Y$ are correlated ...
0
votes
1answer
45 views
Conditional Density, Additive Gaussian
A signal, X, is a random variable with the following density function:
$$f_X(x) =\begin{cases} \frac{3}{25}(x-5)^2, &0 \le x \le 5\\0, &otherwise \end{cases}
$$
The signal is transmitted ...
1
vote
1answer
45 views
maximum of exponentials
I am really having difficulties to prove the following: consider $X_1,\dots, X_n$ all exponentially distributed with rate $\lambda$ (i.e. $X_i \sim exp( 1/\lambda)$). Then argue that we can write
...
0
votes
1answer
56 views
Conditional expectation $E[X|Y<y]$
Let $X:\Omega \to \mathbb{R}$ and $Y:\Omega \to \mathbb{R}$. Consider the joint pdf $f_{XY}(x,y)$ and univariate pdfs $f_X(x)$ and $f_Y(y)$.
Is it true that $E[X|Y < y]$ equals:
$$ \displaystyle ...
1
vote
1answer
38 views
Are the multiplications of i.i.d random variables , i.i.d?
If we know that $X_1$ and $X_2$ are i.i.d random variables, and $Z_1$ and $Z_2$ are also i.i.d random variables, can we say $X_1Z_1$ and $X_2Z_2$ are i.i.d random variables too? suppose that $X_1$ ...
1
vote
0answers
41 views
About Strict Stationary of AR(1) Sequence
The usual Auto regressive process considers the time t from negative infinity and positive infinity, but what if we restrict our time to strict positive space, do we still have our stationary result?
...
1
vote
4answers
54 views
Distribution of a random variable
$X_1$, $X_2$, $X_3$ are independent random variables, each with an exponential distribution, but with means of $2.0, 5.0, 10.0$ respectively. Let $Y$= the smallest or minimum value of these three ...
1
vote
1answer
44 views
Inequality between two Random Walks
Let's consider two Random Walks,
$$x^{(1)}_t = x_0 + \sum_{i=1}^{t}\xi^{(1)}_i,$$
$$x^{(2)}_t = x_0 + \sum_{i=1}^{t}\xi^{(2)}_i.$$
The random variables $\xi^{(1)}_i$ are i. i. d. They take values on ...
-2
votes
1answer
30 views
Statistics Binomial with Probability Distribution function
Let $X$ be a binomial random variable with $n=2, θ=\frac14$. Find the probability distribution function of $Y=(X^2)+2$.
2
votes
1answer
40 views
Independence of two products of random variables
Consider the following problem:
$$z_1 = a_1 x_1$$ $$z_2 = a_2 x_2$$ where $a_1, a_2$ are i.i.d. (regardless of their distribution; in the actual case study it is a symmetric Bernoulli distribution ...
0
votes
0answers
18 views
is there a Kalman filter for distribution function?
The standard Kalman filter uses a series of measurements observed over time, to decomposite the signal and noise.
However, when I'm modeling the distribution (pdf or cdf) of a variant, is there a ...
0
votes
1answer
53 views
How is conditional density function with two given conditions ($f_{X\mid Y,Z}(x\mid y,z)$) defined?
Let $X$, $Y$ and $Z$ be random variables.
Given this conditional density function with two conditions; $Y=y$ and $Z=z$:
$$ f_{X\mid Y,Z}(x \mid y, z) = f_{X\mid Y,Z}(x \mid Y=y, Z=z) $$
I have a ...
0
votes
2answers
35 views
Using join probability distribution
Say I'm given a probability distribution of two random variables $A$ and $B$. What does it mean to calculate the join probability distribution of $3^{(A-B)}$?
The distribution is in fact discrete.
2
votes
1answer
37 views
Do Convergence in Distribution and Convergence of the Variances determine the Variance of the Limit?
Suppose we have a sequence $(X_n)_{n\in\mathbb{N}}$ that satisfies:
$X_n \rightarrow_d X$, for $n\rightarrow \infty$, where $\rightarrow_d$ denotes convergence in distribution;
$\mathrm{Var}(X_n)$ ...
0
votes
2answers
59 views
Cumulative distribution function & expectation
Let a be a real number and f:
$$f:\mathbb{R}\rightarrow \mathbb{R}, \ f(x) = \begin{cases} a3^x & \text{for } x < 0\\ 1& \text{for } x =0 \\ a3^{-x} & \text{for } x > 0\end{cases}$$
...
2
votes
2answers
109 views
conditional distribution of random variable given its sum with another random variable
I am trying to figure out the following problem:
I have two random variables: $X$ with pdf $f_X(x)$ on $[0,A]$ and $Y$ with pdf $g_Y(y)$ on $[0,B]$. Let's denote $Z=X+Y$. What should be the ...
0
votes
1answer
44 views
Problems getting transformation function from source and destination random variables knowledge when handling the discrete case
In this question I asked about a way in order to find a specific transformation function $g(\cdot)$ in order to transform a random variable into another one.
Thanks to the answer to that question I ...
0
votes
2answers
45 views
Expected value given distribution
What would be the variance of a random var. $Z$ with distribution $\mathbb{P}(Z=n)=2^{-n}$ over all positive integers? I am clueless.
I know $\mathbb{E}(Z)$ would be $\sum_{n=1}^\infty n 2^{-n}$. At ...
1
vote
2answers
25 views
r.v Law of the ratio
Earlier the following question was asked, I think by a classmate:
r.v. Law of the min
I posted my solution to it, but I am stumped on the second half of the problem:
If the law of $(X,Y)$ is ...
0
votes
1answer
35 views
Generating log-distributed random variates
I'm looking for a numerically stable way to generate random variates that are distributed like $\log(U)$ with $U \in (0,1)$, to be stored in IEEE 754 floating-point variables. My idea is:
Generate ...
1
vote
1answer
62 views
Weak Convergence to Exponential Random Variable
Assume that $X_1$, $X_2$,... are independent random variables uniformly distributed on $[0,1]$. Let $Y^{(n)}=n\inf\{X_i,1\leq i\leq n\}$. I am asked to prove that it converges weakly to an exponential ...
4
votes
2answers
296 views
Expectation of the min of two independent random variables?
How do you compute the minimum of two independent random variables in the general case ?
In the particular case there would be two uniforms variables with difference support, how should one proceed ?
...
3
votes
3answers
216 views
Repeated convolution of probability distributions
Question
Let $$S_k=\sum_{i=1}^k X_i$$ be the sum of $k$ independent random variables. I am interested in closed-form expressions of the pdf of $S_k$.
In general, the pdf is given by the $k$-fold ...
1
vote
1answer
24 views
N arcs on a ring, which are either blue or red, find expectation and variance of number of red arcs using indicators.
There are n distinct points marked on the ring, each of which is either blue or red
with equal probabilities independently of each other. These n points divide the ring into n arcs.
If an arc has both ...
0
votes
1answer
138 views
sum of two random variables
Can any of you help me?
I have some problem with this exercise of "Probability and Statistics" :
Calculate the probability density function (PDF) of $Z=X+Y$
where $Y$ is discrete random ...
1
vote
2answers
40 views
I want to find the real number c for which we have: $P(X<c)=P(Y<c)$
Consider two normal random variables $X$ and $Y$:
$$X\sim N(m_1,s_1), \qquad
Y\sim N(m_2,s_2)$$
I want to find the real number $c$ for which we have:
$$P(X<c)=P(Y<c)$$
0
votes
2answers
103 views
if X and Y are independent Normal random variables, what would be the probability that X is less than Y?
How can I find this probability $P(X<Y)$ ? knowing that X and Y are independent Normal random variables.
$X$~$N(m_1 ,v_1)$ (with mean m1 and variance v1)
$Y$~$N(m_2 ,v_2)$
I know that
$$ \Pr ...
0
votes
1answer
289 views
Division of two random variables of uniform distributions
Having X ~ Uniform(0,1), Y ~ Uniform(1,3) independent what's the pdf of Z = X/Y.
This means I can write the PDFs as follows $$f_X(x) = 1$$ for $ x \in \left(0,1\right)$ and 0 otherwise
$$f_Y(y) = ...
0
votes
1answer
75 views
Let $F(x,y)=1$ for $x+y\ge 0$ and be zero otherwise. Show that $F$ cannot possibly be the joint distribution function of a pair of random variables.
Let $F(x,y)=1$ for $x+y\geq 0$ and be zero otherwise. Show that $F$ cannot possibly be the joint distribution function of a pair of random variables.
Ok so basically I need to show that there can't ...
3
votes
1answer
61 views
distinguishing two random distributions
Let $x_1,x_2,\ldots,x_n$ and $y_1,y_2,\ldots,y_n$ be uniformly, independent random variables. What can I say about the distribution of the following two variables?
$$S_1 = ...
0
votes
0answers
104 views
what is the pdf of the product of two independent RVs for Normal and chi-square distributed RVs?
what is the pdf of the product of two independent random variables X and Y, if X and Y are independent?
X is normal distributed and Y is chi-square distributed.
Z = XY
if $X$ has normal distribution ...
3
votes
1answer
122 views
Determining boundaries of Probability Density Function integral for a requested probability
This isn't one specific homework question, but a concept I'm having trouble with in class. We were asked on a couple of questions recently on homework dealing with the probability density function of ...
0
votes
1answer
50 views
Poisson distribution and probability distributions
Suppose $X$ has the $\mathrm{Poisson}(5)$ distribution considered earlier.
Then $P(X \in A) = \sum_{j\in A} \frac{e^{-5}5^j}{j!}$, which implies that $L(X) = \sum^\infty_{j=0} ...
1
vote
1answer
19 views
Distributions of random variables
Let $(\omega, F, P)$ be Lebesgue measeure on$[0,1]$, and set
$X(\omega) = 1$ if $0 \le \omega < \frac{1}{4}$
$X(\omega) = 2\omega^2$ if $\frac{1}{4} \le \omega < \frac{3}{4}$
$X(\omega) = ...
1
vote
2answers
93 views
Discrete random variable, joint density question
Discrete random variables $X$ and $Y$ have the joint density shown in the table below.
$ \ \ \ \ X \ 1 \ \ \ 2$
$Y$
$1 \ \ \ \ \ \ \frac{1}{10} \ \ \ \frac{2}{10}$
$2 \ \ \ \ \ \ \frac{3}{10} \ \ ...
1
vote
1answer
53 views
Probability and Stats: 100 random samples, lognormal random variable, probability that average is less than 9?
So we have a lognormal random variable Y with mean 10.2 and standard deviation 15.3. If we take 100 random samples with these properties, what is the probability that the average for this sample is ...
1
vote
0answers
95 views
Independent Exponentially Distributed Random Variables - Athletes Problem??
Q) At a javalin competition two athletes (1 & 2) are competing against each other. Each has one attempt to throw the javalin. Assume the acheived distance of a throw ($L$1 & $L2$) [note these ...
1
vote
1answer
695 views
CDF of sum of dependent random variables
Suppose that $X$ and $Y$ are $dependent$ random variables, what would be the cumulative distribution of $X+Y$?
That is, what is $P(X+Y\le c)$ for any integer c?
Note that we do not know their joint ...
0
votes
1answer
189 views
Cumulative distribution function determine the random variable
I don't know that determine is the right word, but I try to explain. What I need to understand. :) So..
We know's that if a function fit this conditions:
Monotonically non-decreasing for each of its ...
0
votes
1answer
87 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 ...
0
votes
0answers
48 views
How distribution function behaves when $b\to\infty$
Let $X$ be a r.v. with dist func $F$ and let $a < b$ .Find and sketch the distribution function of
$$z=\begin{cases} x & \text{if } |x| \leq b \\
0 & \text{if } |x| > ...
0
votes
1answer
76 views
Find the distribution function of X.
Let the point (u, v) be chosen uniformly from the square 0<=u<=1, 0<=v<=1. Let X be the random variable that assigns to the point (u, v) the number u+v. Find the distribution function of ...
0
votes
1answer
122 views
Convergence of sequence of Bernoulli random variables
I'm stuck with this problem.
Let $X_1, X_2, ...$ be a sequence of independent Bernoulli random variables. Show that if $$\sum_{i=1}^n \frac{p_i}{n} \to l \; \text{ as } \; n \to \infty$$ then ...
1
vote
1answer
70 views
What is this question on random variables asking?
The question states:
A random variable $X$ is called symmetric about 0 if for all $x \in \mathbb R$, $\mathbb P(X \geq x) = \mathbb P(X \leq -x)$.
Prove that if $X$ is symmetric about 0, ...
