1
vote
1answer
19 views

Time Series Analysis.Calculate the variance mean and autocorrelation of the time series below.

For the following time series, calculate the mean, varia nce and autocorrelation function: (a) Y_t=5+Z_t+ 0.6Z_t-1
1
vote
1answer
36 views

Finding the mean with absolute value

This question is out of my field and topic that I am teaching myself now, but I was wondering how would you solve this problem if it had the absolute value of it. My Question: $$f(x) = ...
1
vote
0answers
34 views

the marginal pdf of a transformed variable from a joint distrubution

The questions tells us to let X and Y be random variables for which the joint p.d.f. is as follows: $$f(x,y)= \begin{cases} 2(x+y), & \text{for $0 \le\ y \le\ x \le\ 1$} \\ 0, & ...
1
vote
1answer
42 views

Is the derivative of a continuous density function on $\mathbb{R}$ integrable?

Any continuous density function on $\mathbb{R}$ is integrable. Is it true that all derivatives of a probability density function defined on $\mathbb{R}$ is also integrable?
0
votes
0answers
46 views

Is $(\ln l(y))^2 l(y)^x f_0(y)$ integrable over the real numbers?

Is $(\ln l(y))^2 l(y)^x f_0(y)$ integrable over $\mathbb{R}$ for any continuous pair of densities $f_0$, $f_1$ and $l=f_1/f_0$ with some known constant $0\leq x\leq 1$? It seems that $(\ln ...
1
vote
1answer
30 views

Weird question about probability density function

I'm assuming "actual" means the total probability of the PDF (the integral from $-\infty to \infty$) must be 1, so $$\int\limits_{-\infty}^{\infty} ke^{-0.1t}dt = 1$$ Wolfram Alpha seems to be ...
0
votes
1answer
16 views

Calculating the distribution of the minimum of two exponential functions

Suppose X and Y are two independent exponential random variables with rates $\alpha$ and $\beta$ respectively. I know the following equality to be true but I don't know why it's true: $\mathbb{P}(Y ...
0
votes
0answers
34 views

Integrals as continuous sums

I came a cross this paragraph while reading a book on probability: "The length of a subset $S$ of $[0,1]$ is the integral $\int_S dt$". Unfortunately I am unable to understand this part. Can anyone ...
0
votes
1answer
34 views

Double Integration with interesting variable limits, and difficult function

I am trying to reconstruct the following probabilistic model, \begin{equation} \begin{split} \frac{1}{\mu}\int^{\infty}_{0}P(N \geq n\, |\, L=l, T=t)\,e^{-\frac{l}{\mu}} dl &= ...
0
votes
1answer
11 views

Finding Marginal Density function of a joint density function

$$f(y_1,y2) = (1/2)y_1 + (1/4)y_2,\\ 0 \le y_1 \le 1, \\0 \le y_2 \le 2 \\ 0\text{ elsewhere.} $$ How would I find the marginal density function for $Y_1$, and $Y_2$? To find $Y_1$ you would do ...
1
vote
0answers
38 views

Minimum of N Chi-square random variables when N is large

I have a problem in numerically evaluating the PDF of $Y=\min(X_1,X_2,\cdots,X_N)$ where $N=\binom{M}{K}$, the binomial coefficient and $X_i$s are iid Chi-square random variables. The CDF of $Y$ is ...
1
vote
0answers
29 views

expected value with integration

For the exponential distribution, $f(x)=(1/\theta) e^{-x/\theta}$ for $x>0,$ and $f(x)=0$ for $x \leq0$ $(i)$ Determine the exact value for the probability $P(0<X<3\theta).$ I need help ...
1
vote
1answer
31 views

Probability that there is an edge between two nodes in a random geometric graph

I am new to Random geometric graphs. I have a graph with vertices being generated uniformly over $[0,1]^2$. There is an edge between two vertices if the Euclidean distance between the two vertices is ...
-1
votes
0answers
23 views

Probability of rainfall

Suppose that the annual rainfall in a certain region is a uniform random variable with values ranging from 12 to 15 inches. $(i)$ Find the probability that in a given year the region's rainfall will ...
0
votes
0answers
28 views

Proof convolution formula two stochastic variables

Let's say I have two continuous independent stochastic variables, defined on $(0, \infty)$. With densities: $X_1$ ~ $f_1(t_1), t_1 \in (0, \infty)$ $X_2$ ~ $f_2(t_2), t_2 \in (0, \infty)$ The ...
0
votes
1answer
27 views

Lifetime of exponential variable of a battery

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with parameter $\theta=2$ $($measured in years$)$. Find the probability that a battery of this type ...
1
vote
0answers
30 views

Lifetime of pdf disk

The pdf for the lifetime X, in years, of a Superstuff disk drive is given as follows: $f(x) = \begin{cases} 2/x^2 & \text{for } x\geq2\text{ } \\ 0 & \text{elsewhere} \end{cases}$. ...
1
vote
2answers
92 views

What does integration of g(x)f(x)dx mean with known bounds?

What does $\int_{0}^{a}g(x)f(x)dx$ mean if $f(x)$ is the probability density function of $X$? I do know $\int_{-\infty}^{+\infty}xf(x)dx$ is the expected value of $X$, from Wikipedia. But I am not ...
1
vote
1answer
31 views

Random variable of a store

The weekly profit in thousands of dollars of Miller's Office Supply Store is random variable X whose cdf is given as follows: $F(x)=0$ for $x<0$; $F(x)=(3/32)(2x^2-x^3/3)$ for $0 \leq x \leq 4$; ...
0
votes
1answer
51 views

Derivative of Expected value

Say we have a random variable $X$, with density function $f(x)$, and moment generating function $M(t) = E[e^{tX}]$, and we take the derivative - $M(t) = \frac{d}{dt}E[e^{tX}] = E[\frac{d}{dt}e^{tX}]$ ...
0
votes
1answer
40 views

Statistics: Integration from a joint probability distribution

If the joint probability density of two random variables is given by: $$f(x_1, x_2) = \begin{cases}6e^{-2x_1-3x_2} &\quad \text{for } x_1 > 0,\, x_2 > 0\\ 0,&\quad ...
1
vote
1answer
33 views

Expected value of a random variable $X$ with density function $f(x) = \frac{5}{x^2}$ when $x > 5$

I am looking for $E[X]$ when $X$ has a density function $$f(x) = \frac{5}{x^2}$$ when $x > 5$ and $0$ elsewhere. But $\int_5^\infty x \frac{5}{x^2}dx = 5\int_5^\infty x^{-1}dx = ...
1
vote
1answer
43 views

Compute expectation (Ito integral/calculus)

I am having trouble computing this expectation. Does anyone know how to proceed? $$E\left[e^{2B(t)} \int_0^t s dB(s) \right].$$ Is it 0? I tried expressing $e^{2B(t)}=1+ 2\int_0^t ...
3
votes
1answer
81 views

Integration and theorems on continuous functions

$f(x)$ is positive and continuous function on $\mathbb{R}$ and, moreover, $\int_{-\infty}^{+\infty}f(x)dx=1$. $\alpha\in(0;1)$ and $[a;b]$ is the interval having a minimum length such that the ...
0
votes
0answers
10 views

Existence of invers of a function from covariance matrices space

Let $(\mathbb{R}^k,{\cal A})$ be a measurable space. Fix $c>0$ and for every $X\in \mathbb{R}^k$ define $X_c$ as $k-$dimensional vector such that the $i-th$ element of $X^{(c)}$ is $(-c)\vee ...
0
votes
1answer
32 views

I have some approximate integral calculation. Is there a clean way to prove it?

Let: $P(R)=\int_R^{\infty}F(z)e^{-z}dz$ where $F(z)$ is the CDF of some discreate positive R.V. denote by $U$. Integrate by parts: $P(R)=(-F(z)e^{-z})_R^{\infty}+\int_R^{\infty}F'(z)e^{-z}dz$ The ...
0
votes
1answer
62 views

Integrating with the indicator function of some random variables

I have the following problem. Suppose $X_1, X_2, \cdots, X_{n} $ are independent random variables which are all distributed in $U [ a, b]$. Define a new random variable $$N_{x, y} = \sum_{k=1}^y ...
1
vote
1answer
47 views

Computing the moment-generating function of $f(x)=e^{-x}$

I have the following in my textbook: And I'm trying to verify that $M(t) = (1-t)^{-1}$ on my own. I'm getting: $$ f(x) = e^{-x} $$ $$ M(t) = \int_0^\infty e^{tx}f(x) ~dx $$ $$ = \int_0^\infty ...
1
vote
2answers
33 views

Deduce expected value from conditional probability

If you have: $$\begin{align*} P(x) & = \mathcal N(x\mid\mu_x, \sigma^2_x) \\ P(y\mid x) & = \mathcal N(y\mid ax+b,\sigma^2_y) \end{align*}$$ I want to calculate $E(Y)$. I can see intuitively ...
0
votes
2answers
23 views

Integrating a pdf of a normal distribution is equal to the mean

This is a very simple question I believe. It's not homework; it's part of a module I'm studying that I want to understand better. If you have a random variable $X$ that has a normal distribution. Is ...
2
votes
2answers
75 views

how to solve this integral in survival analysis

Let $T$ be a positive random variable, $S(t)=P(T\geq t)$. Prove that $$E[T]=\int^\infty_0 S(t)dt.$$ I have tried this unsuccessfully.
2
votes
1answer
36 views

What is the value of the following integral? ( including the inverse of the CDF of the standard normal distribution)

What is the value of the following integral? $$f(\alpha)=\int_0^\alpha [b-c \Phi^{-1}(1-\beta)] d\beta$$ where $0<= \alpha=<1$ ,b and c are constants, and $\Phi$ is the cumulative ...
0
votes
1answer
38 views

Probability of setting a new record

I need to determine the probability of setting a new record knowing that the times of the $8$ finalists are $U(9.53;9.6)$, and the record is $9.58$. Thanks! I know I have to use integration, but I ...
0
votes
0answers
158 views

Is the summation of given $3$ integrals always greater than $1$

For two density functions $f_1$ and $f_0$ on $\mathbb{R}$, $l(y)=f_1/f_0(y)$ is an increasing function of $y$. We are also given the following information: Condition ($1$) $\rightarrow$ ...
1
vote
1answer
29 views

Problem in deriving distribution of the product of two floating point numbers

I'm studying the Hamming's article "On the Distribution of Numbers", but i'm confused in the way he obtain the (cumulative) distribution of the product of two floating point numbers. Maybe it's only ...
1
vote
2answers
38 views

Calculate the value of c for which f is a probability density.

Let f the function defined by: Where c is positive none zero and constant . How can i calculate the value of c for which f is a probability density.Thnxs for the help.
1
vote
1answer
33 views

Empirical distribution. Problem with changing variables

We have iid random variables $X_1, X_2, \ldots, X_n$ with continuous cdf $F$. Define empirical distribution function $\hat{F}_n (x)= \frac{1}{n} \sum_{k=1}^n \mathbb{I}_{\{X_i \le x \}}$. Let's ...
3
votes
1answer
106 views

How to arrive at a specific formulation of the relative median deviation? Related to integration and statistics.

So my title is not very specific but here is the question in more detail. I am an economist currently working with this book: Frank Cowell - Measuring Inequality On page 25 a formulation of the ...
5
votes
5answers
107 views

Please explain to me why the Expected Value is $ E[X] = \int_{-\infty}^{\infty} x f_X(x) dx $

For probability density functions (at least for the normal distribution and beta distribution) it holds that the expected value is given by $ E[X] = \int_{-\infty}^{\infty} x f_X(x) \, dx $. I have ...
0
votes
0answers
17 views

Reciprocal antiderivative of a process in an expected value

Given a stochastic process $X=\left \{ X_{t}:t\in [0,T] \right \}$, with known probability and spectral density function, is there a way to calculate or estimate the following expectation: $$\left ...
0
votes
1answer
38 views

Techniques for integrating this function?

I'm working my way through a textbook on probability in which the following integral appears: $$F(y)=\int_1^\infty y^{n-1}\lambda^ne^{-\lambda y}\frac{1}{(n-1)!}dy-\int_1^\infty ...
1
vote
1answer
42 views

Joint To Marginal Density : Can't figure it out.

Here goes the problem: Problem: Suppose $X$ and $Y$ have the joint density function: $f(X,Y) = c \sqrt{1 - x^2 - y^2}, \,\,\,\,\, x^2 + y^2 \leq 1$ Find $c$. ...
0
votes
1answer
141 views

Prove the improper integral of the Gamma function $\Gamma(t)$ converges for $z \in \mathbb C$ with $Re(z) > 0$:

Prove the improper integral of the Gamma function $\Gamma(t)$ converges for $z \in \mathbb C$ with $Re(z) > 0$: The gamma function $\Gamma(t)$ is defined by the following improper integral ...
2
votes
0answers
42 views

Why is the equality true: $\int_{u=0}^1 u^{\alpha_1-1} (1-u)^{\alpha_2-1} \, du =\frac{\Gamma(\alpha_1)\Gamma(\alpha_2)} {\Gamma(\alpha_1+\alpha_2)}$

I have the following equality in a textbook of mine $$\frac{y^{\alpha_1+\alpha_2-1} e^{-y/\beta}}{\Gamma(\alpha_1+\alpha_2) \beta^{\alpha_1+\alpha_2}} \cdot ...
1
vote
0answers
58 views

Adding truncated normals: calculating convolutions

Problem: Suppose that $X$, $Y$, and $Z$ are independent standard normal random variables. What is the probability of: \begin{equation} P\{ X+Y+Z+\Delta>0 \, | \, Z+\Delta>0, \, ...
0
votes
1answer
31 views

Find the value of c

Suppose $x$ has density $f(x) = c/x^4$ for $x > 1$ ($(f(x) = 0$ otherwise) where $ c$ is a constant. Find $c$. *My steps: $\int_{-\infty}^{\infty} f(x) dx =1$ Simplifies to $c\int_{1}^{\infty} ...
0
votes
0answers
32 views

Characteristic function of a exponential random variable, problems with complex integral.

I tried to compute the characteristic function of a random variable, which is exponential distributed with parameter $\lambda$: \begin{align*} \varphi(t) &= \mathbb E[e^{itX}] = ...
1
vote
1answer
38 views

Order of Integration E[X]

I understand that $\int_0^\infty P(X>x)dx=E[x]$, and also the logic behind the discrete version here. What I don't understand is how the limits of integration change as is seen here, from $(x, ...
0
votes
1answer
25 views

Does this inequality hold for a Lipschitzian strictly decreasing function?

There is a function $f : [0;1]^n\rightarrow \mathbb{R}$ which is Lipschitzian and strictly decreasing in all variables. Is it possible to prove either one of these two statements? There is an $M > ...
1
vote
1answer
48 views

How do you find the distribution of this sum?

If $X\sim \text{Normal}(\mu=0, \sigma^2), Y\sim \text{Unif}(0,\pi)$, and $X \perp Y$, how do you find the distribution of $Z=X+a\cdot cos(Y)$ for some $a > 0$ ? I've found the distribution ...