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

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3
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2answers
35 views

Probability/Statistics

Let $\{X_r : r\ge 1\}$ be independently and uniformly distributed on $[0,1]$. Let $0<x<1$ and define $$N=\min\{n\ge 1 : X_1 + X_2 +\ldots+X_n> x\}$$ Show that $$P(N>n) = ...
-1
votes
0answers
19 views

Create the most 'stressful' tennis game ever!

Some games, such as tennis, use a complicated points system (point, game, set, match; with deuces and tie-breaks) for what would otherwise be an extremely simple and monotonous game. The main reason, ...
0
votes
0answers
18 views

Determine whether the following expression is positive

I am faced with a problem where I need to show that an expected expression is positive. But I fail to give a strict proof. $$A=E_v ...
0
votes
0answers
8 views

PDF of product of two continous joint distribution

Suppose that $X1$ and $X2$ have a continuous joint distribution for which the joint PDF is as follows: \begin{equation*} f(x_1,x_2) = \begin{cases} x_1 + x_2 & \text {for $0 < x_1 < ...
0
votes
0answers
10 views

Find all random variables whose distribution satisfies an equation

The problem I have to solve is formulated as follows: Find all random variables such that if $Y$ has the distribution $N(0,1)$ and $X, Y$ are independent then $X+Y$ has the same distribution as ...
0
votes
1answer
14 views

Conditional probability density function

Let $\theta$ be the parameter of the probability density function $f(x)$. If it is mentioned that $f(x|\theta)$ be the conditional probability density function, then what does $f(x|\theta)$ mean? ...
2
votes
2answers
93 views
+50

$E_n =\lbrace X_n > X_m \ \forall m < n \rbrace $ are independent

I'm stuck with this exercise. Suppose $(X_n)$ are independent random variables defined on $(\Omega, \mathfrak{F}, P)$ with the same p.d.f. Let $E_1 = \Omega$ and for $n \geq 2$ $$E_n =\lbrace X_n ...
2
votes
1answer
13 views

Conditional expectation of $Y_1$ given that $\sup Y_i=z$, for $(Y_i)$ i.i.d. uniform on $[0,\theta]$

Suppose that $Y_1,\ldots,Y_n$ are random variables independently and identically distributed as uniform on $[0,\theta]$ for some $\theta>0$. How do I find the conditional density of $Y_1$ given ...
0
votes
1answer
16 views

Please help me with the solution of the following problem:

Each bag in a large box contains 25 tulip bulbs. It is known that 60% of the bags contain bulbs for 5 red and 20 yellow tulips, while the remaining 40% of the bags contain bulbs for 15 red and 10 ...
1
vote
1answer
24 views

Whats the formula for the probability density function of skewed normal distribution

The formula for the probability density function of a standard normal distribution that isn't skewed is: $$P(x) = \frac{1}{\sqrt{2π}}e^{-(x^2 / 2)}$$ where, $π = 3.14, e = 2.718$. What if it is ...
2
votes
2answers
41 views

Find $E(|X-Y|^a)$ where $X$ and $Y$ are independent uniform on $(0,1)$

Let $X,Y$ be independent $Uniform(0,1)$ random variables. Find $E(|X-Y|^a)$ where $a>0$. My working: Define $W=1$ if $X>Y$ and $W=0$ if $X<Y$. We seek ...
1
vote
2answers
386 views

PDF and CDF of the division of two Random variables

I have two RVs; their PDF are as the followings: \begin{split} f_{X}(x) = \frac 1 {a} e^{-\frac x {a}}\end{split} and \begin{split} f_{Y}(y) = \frac {y^{L-1}} {b^{L} \Gamma (L)} e^{-\frac y ...
0
votes
2answers
289 views

Joint distribution of U = X + Y and V = X - Y

I have two independent continuous random variables, X and Y, which are uniformly distributed over the interval [0,1]. From this I have two further random variables, U and V, which are defined as U = X ...
1
vote
1answer
22 views

Let X and Y be geometrically distributed iid r.v.s. Find the pmf of min(X, Y), and the pmf and Z = X - Y.

Let X and Y be geometrically distributed iid r.v.s. Find the pmf of M = min(X, Y), and the pmf and D = X - Y. I thought $$ P(M = m) = P(X = x) \cdot P(Y > x) + P(Y = y) \cdot P(X > y) + ...
0
votes
1answer
47 views

Explicit CDF associated to Gamma PDF

Let the distribution function of $X$ for $x>0$ be: $$F(x) = 1 - \sum_0^3 \frac{x^ke^{-x}}{k!}$$ what is the density function of $X$ for $x > 0$? This is what I'm thinking: $$ ...
0
votes
0answers
18 views

Calculating Power of a Paired T Test

$ 239$ subjects had their cholesterol measured, and then were put on high-fiber diets. After a month on the high-fiber diet, the cholesterol was measured again. The mean LDL cholesterol level before ...
-1
votes
0answers
19 views

Repeated coin flips probability [on hold]

Assume in an experiment, one flips a coin $L$ times. This experiment is repeated T times. Assume the $k$'th flip for all possible $k$ values ($1 \le k \le L$) among all experiements. If the head ...
0
votes
3answers
30 views

How can I calculte the probability of $X$ with a Generlized Hyperbolic Distribution?

I would like to know how to calculate the probability of $X$ when I have fitted a Generalized Hyperbolic Distribution to my data set. The depth of my knowledge is basic t-tests and z-tests. I am ...
2
votes
0answers
29 views

Can anything be learned about a probability distribution *directly* from its characteristic function?

Some preliminaries: I know that one can take the inverse Fourier transform to get back the pdf...that is not what I am after. My question is whether the characteristic function, qua function, tells us ...
0
votes
1answer
11 views

What is the probability that $x$ will not work due to failure rate $0.0111$

I've tried using the probability mass function for binomial distribution in this case but it seems to not be the appropriate approach unless I calculated wrong. How am I supposed to approach this ...
3
votes
1answer
40 views

Is my method of working fine?

Suppose a point $X$ is selected at random from a line segment $AB$ of length $l$ and midpoint $O$. Find the probability that $AX,BX$ and $AO$ form a triangle. My method and working is: Case ...
1
vote
1answer
345 views

How to prove that the binomial distribution is approximately close to the normal distribution when $np(1-p) \geq 10$

I would like a formal proof for this "rule of thumb." Can you assist me in getting to this solution? I require the insights and creativities of mathematicians. We know that if $np(1-p) \geq 10$ the ...
-2
votes
0answers
27 views

conditional probability proof 3 varables [on hold]

Suppose that $\mathcal a$ ,$\mathcal b$ and $\mathcal c$ are dependent variables. $$\mathbb P(a \mid b) = \sum \mathbb P(a \mid b,c) \ \mathbb P(c \mid b)$$ can anyone explain it how we get it?
0
votes
0answers
21 views

How to use joint probability density to check for independent events?

Suppose that the joint PDF of $X$ and $Y$ is as follows: $$ f(x) = \begin{cases} 24xy & \text {$x \geq 0, y \geq 0, x+y \leq 1$}\\ 0 & \text {otherwise ...
5
votes
1answer
34 views

Not getting the answer as given in Feller

Find the probability that the equation $x^2-2ax+b=0$ has complex roots, if $a,b$ are random variables following the Uniform $(0,h)$ distribution individually and independently. So we effectively ...
1
vote
1answer
128 views

Markov's Inequality for Negative Binomial distribution

Given that $Y$ follows Negative Binomial distribution (counts y successes before $k$th failure), using Markov's inequality show that for any $q \in[p,1]$, there exists constant $C$, such that ...
8
votes
1answer
397 views

Volume of the intersection of ellipsoids

How do I compute the volume of the intersection of two $n$-dimensional ellipsoids? Given an $n$-vector $c$ and a symmetric positive-definite $n\times n$ matrix $A$, define the ellipsoid ...
2
votes
1answer
38 views

Calculate the mean, the median and the quartiles.

Let $D=\{(x,y):x>0,x^2+y^2<1\}$ and let $(X,Y)$ be the random variable with the density: $$f(x,y)=\frac{2}{\pi}1_{D}(x,y).$$ Let $Z=\frac{Y}{X}$. Calculate the mean, the median and the first and ...
2
votes
2answers
27 views

Convergence in law of sample means of random variable

Let $\{X_n | n \in \mathbb{N} \}$ be a sequence of independent identically distributed random variables with density function: $$f_X(x) = e^{\theta - x}I_{(\theta, \infty)}(x)$$ with $\theta > ...
0
votes
1answer
30 views

Calculate the probability given by three random variables

Let $X_1,X_2,X_3$ be IID random variables, each with the density $$f(x)=x e^{-x}\cdot 1_{(0,\infty)}(x).$$ Calculate $P(X_1+X_2+X_3>4,X_1+X_2<4)$.
0
votes
0answers
10 views

Obtaining the Log-logistic distribution from a truncated logistic distribution

Let $$f(x) = \frac{e^x}{(1+e^x)^2}~,~ -\infty \lt x \lt \infty~~~~~(1)$$ be the standard logistic pdf of a random variable $X$. Then one can obtain the pdf of the log-logistic distribution via the ...
0
votes
0answers
17 views

Show that for any geometric random variable $X$ and parameter $p, \mathrm{Pr}(X < t) = 1 − p^t$. [on hold]

How to prove the above stated equation? I tried the following : Pr⁡(X(i=1)^(t-1)▒〖Pr⁡(X=i)〗 =∑(i=1)^(t-1)▒〖p(1-p)i-1〗 =1-(1-p)t-1
1
vote
0answers
125 views

Probability density function of $A = B + C$ via Joint Characteristic function of $B$ and $C$

This problem is actually a subproblem of a longer derivation that I am trying to understand. I hope that I striped away all the unnecessary stuff that is not relevant. Please correct me if the ...
1
vote
1answer
486 views

probability distribution $X, Y$ and $X+Y$

A box contains $5$ ticket, $\{ 0 , 0 , 0 , 4 , 4\}$. Drawing two tickets at random w/o replacement. $X$ be the sum of the first two draws and Y be the outcome of the first draw. Question: Find ...
0
votes
1answer
432 views

Use the MGF (Moment Generating Function) to find the joint distribution of $X$ and $Y$

Let $V$ and $W$ be independent standard normal random variables where $X=V+W$ and $Y=3W$ This is what I did: $$M_{x,y}(s,t)=E(e^{sx+ty})=E(e^{s(v+w)+t(3w)})=E(e^{sv+sw+3tw})=E(e^{vs+w(s+3t)})$$ ...
0
votes
0answers
28 views

How to demonstrate the pdf of $P_{\sigma} (t)=\lambda_c e^{- \lambda_c t} / (1 - e^{- \lambda_c T})$

In $t_c$, there are $n$ expirations of $T$ and the remnant $\sigma$ seen from the above figure. Let the time $t_c$ forms the exponential distribution with parameter $\lambda_c$. How to demonstrate ...
0
votes
0answers
12 views

To sketch a “typical” plot of a specific time series model

Let X have a distribution with mean $\mu$ and variance $\sigma^2$, and let $Y_t = X$ for all t. Sketch a “typical” time plot of $Y_t$. My thoughts: This process $Y_t$ is stationary with mean $\mu$, ...
0
votes
0answers
32 views

ODE for the normal distribution [on hold]

The normal density function $\phi(x)=\tfrac{1}{2\pi}e^{-\frac{x^2}2}$ can be described via the ODE $$\phi^\prime(x) = -x \phi(x)$$ under the condition $\int_{-\infty}^\infty \phi(x) = 1$. Is there ...
1
vote
0answers
22 views

Find the moment generation function of $Y=1-e^{-X}$. [on hold]

If $X$ is random variable with PDF: $f(x)=e^{-x}$, $x>0$. Then find the moment generating function of $Y=1-e^{-X}$. Okay, so I don't get why $Y$ is equated to small $x$ since $Y$ itself is a ...
1
vote
1answer
35 views

A Seemingly Trivial but Computationally Complicated Probability Problem

Suppose $X,Y$ are independent $Uniform(-1,1)$ random variables. Determine the distribution of $Z=X-Y$. I do not really think I should add my work here because whatever I have tried until now, has ...
0
votes
0answers
7 views

Sufficient condition for not having infinitely small modes in a distribution

I was reading the paper Optimal Throughput and Delay in Delay-tolerant Networks with Ballistic Mobility (http://dl.acm.org/citation.cfm?id=2500432), and found the following proposition (page 305): ...
2
votes
2answers
925 views

Expected Value of Normal CDF

I am trying to calculate the expected value of a Normal CDF, but I have gotten stuck. I want to find the expected value of $\Phi( \frac{a-bX}{c} )$ where $X$ is distributed as $\mathcal{N}(0,1)$ and ...
7
votes
1answer
744 views

Median of the F-distribution

Is the median of the F-distribution with m and n degrees of freedom decreasing in n, for any m? From experiments it looks like it might be, but I have been unable to prove it.
1
vote
2answers
53 views

Deriving a joint cdf from a joint pdf

I see that a similar question was asked last year, but I am still confused. I have $f(x,y) = 2e^{-x-y}$, $ 0 < x < y < \infty $ and need to find the joint CDF. I have a solution that ...
-1
votes
2answers
41 views

Help me understand how to take derivative of the PDF of X~binom(n,p) with respect to p.

This is the solution I was given. My questions: Why is it summed from k=1 to x. Shouldn't it be from k=1 to n? (If not, why not?) What is happening to the first term from line 1 to line 2? When we ...
0
votes
2answers
416 views

Mean and covariance of Wiener process

Let $x(t), x(0)=0$ be a Wiener process with the parameters $a$ and $\sigma.$ Prove that its mean equals $a \cdot t$ and its covariance $R(t,s)$ is equal $R(t,s)=\sigma \min(t,s)$
1
vote
3answers
40 views

Probability of number of people in car park at any given time

A building has 22 car spaces, each having a car parked within each spot in the morning. Each car is retrieved by its respective owner at some point (random time) between 7am and 9am (120minutes). Each ...
0
votes
1answer
31 views

Comparing sums of random variables

Consider $X_0,X_1\ldots,X_n$ mutually independent and $X_i \sim U(a_i,b_i)$. What is the probability that $\sum_{i=1}^n X_i<X_0$? Can you extend to mutually independent random variables with ...
-7
votes
2answers
80 views

Comparing uniform random variables.

$X$ is a uniformly distributed random variable on $(0,1)$ $Y$ is a uniformly distributed random variable on $(0,2)$ $Z$ is a uniformly distributed random variable on $(0,4)$ What is the probability ...
0
votes
2answers
57 views

Fitting probability distributions based on moment generating functions

Say I have a random variable $X$ with mgf $M_X(t) = 1 + a_1t + a_2t^2 + a_3t^3 + \cdots $ and another random variable $Y$ with a probability distribution determined by two parameters $\theta_1$ and ...