0
votes
2answers
30 views

Probability that the call will be answered at time $t$ is given by $f(t)$. Find the median waiting time for the call.

$$f(t) = \begin{cases} 0 & \text{if $t < 0$ } \\ 0.2e^{-t/5} & \text{if $t\geq 0$} \end{cases}$$. $ $ Find the median waiting time for the call. $ $ I cannot understand ...
0
votes
1answer
22 views

Finding the boundaries of integration when calculating P(X + Y > a) or P(X + Y < b) (Jointly Distributed Continuous Random Variables)

I have a problem on setting the boundaries of integration when I'm trying to find probabilities like $P(X + Y > a)$ or $P(X + Y < b)$. For example, when I have $f(x,y) = \frac {x} {5}\ +\frac ...
2
votes
1answer
77 views

Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,\frac{t}{d})$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function. Note that I have the Laplace transform of : ...
0
votes
0answers
36 views

Urgent Find the CDF of U [duplicate]

I am having problems with Part 2. I know the upper limit of Y is x+u in the formula. But what about the limits of X ? Please help me !!
1
vote
1answer
38 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) = ...
0
votes
2answers
18 views

Area under the PDF of an order statitics

Consider a continuous random variable $$X=\min\{Y_1,Y_2,Y_3\}$$ where $Y_1,Y_2,Y_3$ are iid, non-negative random variables having the same PDF, $f_{Y}(x)$ and CDF $F_{Y}(x)$. The PDF of X is ...
1
vote
1answer
45 views

Problem understanding generation of a dataset within some interval and probability

(1) Generate a dataset within interval $I =[−3,3]×[−3,3]∈\mathbb R^2$ a set $S$ containing $M=10,000$ data points. A point $(x,y)∈I$ is to belong with probability $p(x,y) = p_x(x) p_y(y)$ to ...
0
votes
1answer
24 views

Problem of integral for the marginal density function

It is given the joint probability density function of two random variables X and Y \begin{equation*} f(x,y) = \begin{cases} \frac{\pi \ln(2)}{1+x^2+y^2} & \text{if } x^2 + y^2 \le 1,\\ 0 & ...
2
votes
1answer
48 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
40 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
49 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 $c$. ...
1
vote
2answers
130 views

Integrating a special skew normal — the CDF of a convolution of a normal with a truncated normal

I am having a little trouble trying to compute an integral. In short, I wish to solve the following: $$F(x) = \int_{-\infty}^x \phi(au-b)\,\Phi(au+b)\,du $$ My intuition is that this might be ...
0
votes
3answers
64 views

Fast way to do this well-known integral (gaussian-distribution)

I want to evaluate $$ \frac{1}{\sqrt{2 \pi } \sigma}\int_{-\infty}^{\infty} x^2e^{-\frac{(x-\mu)^2}{2\sigma ^2}}dx.$$ The problem is, I don't want to run into heavy calculations. Therefore, maybe ...
1
vote
1answer
28 views
0
votes
0answers
84 views

Integrating of the product of the Complementary Error Function (Erfc) and Exponential and other variables with unknown power.

I want to evaluate or simplify or (getting closed form expression) the following equation: $$ \begin{equation}error ~probability= \frac{a}{2}\int_0^\infty \mathrm {erfc}\left( {\sqrt\frac{b\cdot ...
2
votes
1answer
50 views

How $\sum_{r=m}^{\infty}\frac{e^{-\lambda}\lambda^r}{r!}=\int_{0}^{\lambda}\frac{e^{-u}u^{m-1}}{(m-1)!}du$

$$P(X\geq m)=\sum_{r=m}^{\infty}\frac{e^{-\lambda}\lambda^r}{r!};m=0,1,...$$ Show that for any $m=1,2,...$ $$P(X\geq m)=\int_{0}^{\lambda}\frac{e^{-u}u^{m-1}}{(m-1)!}du$$ I couldn't derive it also ...
0
votes
1answer
43 views

Find $\mathbb E[(X+1)^2]$

The continuous random variable $X$ has the following probability density function (pdf), $$f(x) =\frac{3}{(1 + x)^3} ; 0 ≤ x ≤\sqrt3-1$$ I have to find $\mathbb E[(X+1)^2]$ $$\mathbb ...
0
votes
1answer
46 views

Marginal distribution of $P$

Joint distribution of $P$ & $Q$ is $$f_{P,Q}(p,q)=\frac{1}{2\sqrt{(2\pi)}\sigma}\exp[-\frac{1}{2}{(\frac{\frac{p+q}{2}-\mu}{\sigma})^2}] \times\theta\exp[-\theta(\frac{p-q}{2})],\quad ...
3
votes
2answers
362 views

Characteristic Function of Inverse Gaussian Distribution

The pdf of Inverse Gaussian distribution, IG$(\mu,\lambda)$, is : $$p_X(x)=\sqrt\frac{\lambda}{2\pi x^3}\exp\left[\frac{-\lambda}{2\mu^2x}(x-\mu)^2\right];\quad x>0,\lambda,\mu>0$$ I have to ...
0
votes
1answer
799 views

Cumulative Distribution Function of Logistic Distribution.

pdf of logistic distribution is : $$p_X(x)=\frac{\pi}{\sigma\sqrt 3}\frac{\exp[\frac{-\pi(x-\mu)}{\sigma\sqrt3}]}{(1+\exp[\frac{-\pi(x-\mu)}{\sigma\sqrt3}])^2};\quad-\infty<x<\infty$$ I have ...
7
votes
3answers
2k views

Derivation of the density function of student t-distribution from this big integral.

My lecturer posed a question where we derive the density function of the student t-distribution from the Chi-square and Standard normal distribution. I worked on this question for days, and I am ...
3
votes
1answer
279 views

Integral of product of normal cdf and pdf

What do you think, is there a closed form solution of the following Integral $\textbf{ }$ $$\int_{-\infty}^{a-y}n(x)\, N(b-2y-x)\, dx,$$ where $N(x)=\int_{-\infty}^x n(z)\, dz\quad$ and $\quad ...
2
votes
1answer
562 views

How to find limits of integration on a convolution of CRVs

In finding the convolution of two independent and continuous random variables, I am struggling with limits of integration. I cannot seem to figure out over what intervals the probability density ...
2
votes
1answer
217 views

Covariance of Student's t-distribution

Another integral (this time it looks like a lot of work but maybe it can be simplified). I have the Student's t-distribution $$\int_{-\infty}^\infty ...
2
votes
1answer
2k views

Taking the derivative of definite integral?

I'm having trouble understanding the derivative of definite integral. For example, why is the following true? $\frac{d}{dx}\displaystyle\int_{0}^{x}F_{1}(x-v)f_{1}(v)\, \mathrm{d}v = ...
2
votes
1answer
345 views

Computing an integral involving standard normal pdf and cdf - with peculiar limits.

I have had a look at some of the other questions on this topic but cannot quite work out the solution to this integral (or prove that there isn't a solution). Is there a way to work out: ...
1
vote
1answer
71 views

Transforming a Continuous Function

My math is quite limited so please bear with me. I will get to the point: Is there a way to transform a continuous function into a bounded one? In essence I have a normalized Gaussian distribution ...
1
vote
2answers
144 views

Question on the standard normal distribution.

Let $X$ be a random variable having standard normal distribution. Let $\Phi$ denote its distribution function. Find $$ \int_0^\infty \operatorname{Prob} (\Phi(X) \geq u) \; du $$
23
votes
7answers
2k views

Prove: $\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx$ for $0 \leq k \leq n$

I would like your help with proving that for every $0 \leq k \leq n$, $$\binom{n}{k}^{-1}=(n+1)\int_{0}^{1}x^{k}(1-x)^{n-k}dx . $$ I tried to integration by parts and to get a pattern or to ...