1
vote
3answers
40 views

Compute variance, using explicit PDF

I'm trying to get $\text{Var}(x)$ of $f(x) = 2(1+x)^{-3},\ x>0$. Please tell me if my working is correct and/or whether there is a better method I can use to get this more easily. $$ ...
0
votes
2answers
33 views

Help finding k. Issue with integration

Let the continuous random variable $X$ have a probability density function $f(x)$ such that $$f(x) = k(1+x)^{-3}, x>0$$ $=0$ elsewhere Find k This is what I tried: $\int_0^\infty k(1+x)^{-3}dx ...
-1
votes
1answer
31 views

Cumulative distribution function [closed]

The delay of the train in minutes is given by the CDF $$F(x) =\begin{cases} \dfrac{(x+5)}{30} &;\ -5<x<0, \\[15pt] \dfrac{2}{3} + \dfrac{x}{180} &;\ 0<x<60. \end{cases} $$ ...
4
votes
3answers
78 views

Deriving Mean and Variance of Laplace Distribution

It has been a long time since I have used calculus, and I am trying to understand how the mean and variance of the Laplace distribution with pdf $$f(x|\mu,\sigma) = \dfrac{1}{2 ...
3
votes
1answer
53 views

Calculus Question: Improper integral $\displaystyle\int_{-\infty}^{\infty} x^{2}e^{x-e^{2x}}dx$

I am curious about evaluation of the following integral $$\int_{-\infty}^{\infty} x^{2}e^{x-e^{2x}}dx$$ Is it possible to evaluate it? This not my homework but I will share my attempt. I tried ...
2
votes
2answers
77 views

Log normal distribution - Where am I wrong?

Let $X$ be a R.V whose pdf is given by $$f(x)=p\frac{1}{\sqrt{2\pi\sigma_1^2}}\exp\left(-\frac{(x-\mu_1)^2}{2\sigma_1^2}\right)+ ...
0
votes
1answer
40 views

Determine the target weight so that no more than 5% of boxes with normal weight distribution contain less than 500 g [closed]

Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g. Suppose a law states that no more than 5% ...
1
vote
0answers
20 views

Integrating with indicator functions

I want to evaluate $$\int_{-\infty}^{\infty}(A_1e^{-\beta_1(b-x-y)}+B_1e^{-\beta_2(b-x-y)})(pn_1e^{-n_1y}1_{\{y\geq0\}}+qn_2e^{n_2y}1_{\{y<0\}})dy,$$ $b>x, \beta_1<n<\beta_2$. I am trying ...
2
votes
1answer
34 views

derivation law from the call option formula

i am reading a article about the option pricing. and i got stuck with some typical statement. $C(K)=\int (x-K)^+\mu(dx)$ is given. here, $\mu$ is implied law of asset price and C(K) is the price ...
0
votes
1answer
21 views

Infimum of Gamma distribution

Let $X$ be a Gamma random variable with the CDF $F_X(x)=\frac{1}{\Gamma(\alpha)}\gamma(\alpha,\beta x)$ where $\Gamma(x)$ represent the gamma function and $\gamma(a,b)$ denotes the lower-incomplete ...
1
vote
1answer
56 views

What distribution is this?

Top: Uniform, Bottom: ?? Distribution. Ignore the random spikes - those are just binning errors. Looking for a distribution that is on $[0,1]$ and is equal to $0$ at $1$ and some positive ...
0
votes
2answers
37 views

How was this integral set up to compute $Pr(X+Y) \geq\frac{\pi}{2}$?

I am trying to understand how to deal with the following type of question given two random variables $X$ and $Y$ that are jointly continuous with some pdf: Here: $f_{X,Y}(x,y) = \left\{ \begin{align} ...
0
votes
0answers
30 views

Change of variables in calculating the integral of multivariable differential entropy

I know that for one dimensional differential entropy of a density function $p(x)$, one has the following formula by change of variables: $$H(p)=\int ...
1
vote
1answer
68 views

Evenly spreading study over 20 days?

Want to study $8$ hours a day, and $4$ hours on exam days. Want to study all exams exactly the same amount of time total. Exams in $10$,$13$,$19$,$20$ days from start. What is the daily ...
0
votes
2answers
35 views

gamma function question relating to normal distribution

I'm trying to show that $\Gamma(1/2)=\sqrt\pi$. A hint I've been given is to use a change of variable and then relate it to normal distribution density. However, I'm really confused as to how I would ...
1
vote
1answer
32 views

How is this step completed?

User Did, did this step in his answer to my previous question: $$\sum_{k=0}^n{n\choose k}(zp)^kq^{n-k}=(q+pz)^n.$$ How is it done? Is it simply an identity, or something more?
2
votes
1answer
164 views

Sum of two gamma/Erlang random variables $\Gamma(m,\lambda)$ and $\Gamma(n, \mu)$ with integer numbers $m \neq n, \lambda \neq \mu$

The gamma distribution with parameters $m > 0$ and $\lambda > 0$ (denoted $\Gamma(m, \lambda)$) has density function $$f(x) = \frac{\lambda e^{-\lambda x} (\lambda x)^{m - 1}}{\Gamma(m)}, x > ...
1
vote
3answers
53 views

Chebyshev inequality for $n=1$?

Wikipedia suggests that Chebyhev's inequality is only true for $n \ge 2$, but I don't see why we have to exclude the case $n=1$? Is wikipedia right? Chebyshev
0
votes
0answers
309 views

How can I derive the PDF from conditional probabilities?

I have some function $P(i)$ which is the probability of success for an experiment on the $i$th trial. The probability mass function for the first successful trial is: $$PMF(n) = \left( ...
0
votes
0answers
27 views

How do I convert a Gamma-distributed random variable, $ \Gamma(2,1)$ to Erlang distribution?

I know that a Gamma-distributed random variable with $\alpha$ as an integer can be converted to Erlang distribution but how? and how do I write it's new probability density function?
0
votes
1answer
29 views

systematic way of finding the bounds for change of variables (multivariable case), Jacobian

Let's say that $X,Y$ are independent standard normal random variables. I am interested in the distribution $P(X+Y\le 2t)$. Clearly, the domain of integration in this case is $-\infty<x<\infty$ ...
1
vote
1answer
34 views

Finding conditionally expected $y$ given a specific $x$ from a joint distribution function!

I want to determine expected $y$, given $x=2$ given joint pdf shown below $$\frac{1}{2y} * e^{-\frac{y^2 + \frac{x}{2}}{y}}$$ for $x,y \gt 0$ and $0$ otherwise I believe this means I want ...
1
vote
1answer
39 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) = ...
3
votes
1answer
32 views

Maximum likelihood to throw exactly two 6s

One throws a dice $n$ times. For which value of $n$ is maximum the probability to obtain exactly two 6s? I get $$n=11 \text{ or } n=12.$$ My solution: the probability to obtain exactly two 6s in ...
1
vote
1answer
35 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 ...
1
vote
0answers
42 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
34 views

Addition corresponds to convolution and subtraction?

We know that if two random variables have proper densities, than the density of the sum of them is given by the convolution. But what can we say about the difference of two random variables? $X-Y$ ...
1
vote
0answers
26 views

Battery between liftimes

Suppose that the operating lifetime of a certain type of battery is an exponential random variable with $\theta$ $= 2$ (measured in years). Find the probability that a battery of this type will have ...
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
78 views

Distribution of ceiling function and absolute value of random variable

Given a distribution function $f_X$, where $X$ is some random variable. I want to get the distribution functions of $|X|$ and $\lceil X \rceil$( the last one may only have an easy form if $X$ is ...
0
votes
2answers
332 views

Version 2:Help finding the probability that $Ax^2 + Bx + C$ has real roots?

Suppose that $A, B,$ and $C$ are independent random variables, each being uniformly distributed over $(0,1)$. What is the probability that $AX^2 + BX + C$ has real roots? I am given a hint that if ...
0
votes
1answer
33 views

Check for Independence

Given $$f_{(U_1,U_2)}(u_1,u_2)=\begin{cases} 1/2& -u_1<u_2<u_1 \text{ and } u_1 - 2 < u_2 < 2 - u_1 \text{ and } 0 < u_1 <2\\ 0& \text{otherwise}\end{cases}$$ I found that ...
0
votes
0answers
76 views

If $X_1, X_2$ have exponential distributions what distribution does $Y = \sqrt{X_1^2 + X_2^2}$ have?

If $P_{X_1}(x) = P_{X_2}(x) = k \exp(-k x)$ how will $Y = \sqrt{X_1^2 + X_2^2}$ be distributed? $X_1$ and $X_2$ are independent. What I have Done: The distribution for $Y=X_i^2$ must be ...
1
vote
1answer
92 views

Multiplying two Gamma distributions over the same variable

I am looking at a software library where there is a function that multiplies two Gamma distributions defined over the same random variable. So, it is basically multiplying two Gamma densities with ...
2
votes
1answer
58 views

Deriving statistical distributions from games

The normal distribution can be derived from basic principles and calculus The Normal Distribution: A derivation from basic principles. Are there other distributions that can be derived like this from ...
1
vote
2answers
42 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.
0
votes
1answer
25 views

Zisserman Lecture and $x_{MLE}$

In the Zisserman Lecture below http://www.robots.ox.ac.uk/~az/lectures/est/lect34.pdf page 36, he derives $x_{MLE}$ for Gaussian sensor fusion. There are two noisy measurements $z_1$ and $z_2$ ...
3
votes
1answer
488 views

Is this infinite sum always less than zero?(+500pts bounty for the correct answer)

I wonder if the following infinite sum is always negative for all (finite) $A,d>0$ and $B<0$. Any counterexample also suffice. Here is the sum: $$\frac{\partial}{\partial d}\sum_{n=1}^\infty n ...
3
votes
1answer
108 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 ...
3
votes
1answer
86 views

How to find the CDF of the binomial distribution in terms of an integral

This wiki page says that the CDF of the binomial distribution in terms of the beta function can be expressed as $$F(k;n,p)=Pr(X\leq k)=(n-k){{n}\choose{k}}\int_0^{1-p}t^{n-k-1}(1-t)^k {d}t$$ How to ...
2
votes
1answer
54 views

Median and Mean of Sum of Two Exponentials

I have a cumulative distribution function: $$G(x) = -ae^{-xb} - ce^{-xd}+h$$ The associated probability density function is: $$g(x) = abe^{-xb} + cde^{-xd}$$ My problem concerns $x\ge 0, X \in R$. I ...
1
vote
1answer
53 views

Normally distributed with probability

Assume the length of waiting at supermarket is approximately normally distributed with mean 6 minutes and standard deviation 1.5 minutes. (1) Fund the probability that waiting time is longer than 8 ...
3
votes
1answer
55 views

Find $a_{n+1}=\frac{a_n^2+1}{2}$ in terms of $n$.

I was trying to prove that for all $n\in \Bbb N$ there are integer numbers $\{a_1,a_2,\ldots,a_n,b_n\}$ s.t. $a_1^2+a_2^2+\dots+a_n^2=b_n^2$. I founded that if $\{a_1,a_2,\ldots,a_n,b_n\}$ have the ...
2
votes
1answer
44 views

Confusion between probability density and probability in EM-paper

I'm reading about expectation maximization from Dempster et al. and there is one point in the paper I get confused about probability density and probability. Maybe you can clarify this to me. Here is ...
2
votes
2answers
136 views

Inner Product vs. Integrals with Fourier Series, When to include 1/2pi?

I am confused about when to include a prefactor of $\frac{1}{2\pi}$ when dealing with integrals of functions that are expressed as fourier series. This is what I understand (please correct me if I'm ...
0
votes
1answer
51 views

Integrals of probability density functions and their inter-relationships

I am a little bit confused about the relationship between marginal probability density functions (pdfs), joint pdfs (jpdfs), and conditional pdfs (cpdfs), and their integrals. Let me define the ...
0
votes
0answers
25 views

Finding a (tighter) sufficient condition on the standard deviation of a random variable

Let $\tilde{\phi}$ be a non-negative random variable with a mean normalized to $1$, with $F(\phi) := \Pr(\tilde{\phi} \leq \phi)$ denoting its CDF. $F(\phi)$ is assumed to be twice continuousy ...
0
votes
1answer
46 views

Where is the error? Expectation, independent random variables

Let $X,Z$ be two correlated variables and $Y,Z\sim N(0,1)$ where Y is independent of $X,Z$. Consider the expectation: $$E[f(X,Y)Z].$$ If $f(X,Y)$ and $Z$ are independent then clearly ...
0
votes
0answers
31 views

Bounding the standard deviation of a random variable

I have the following problem. Let $\tilde{\theta}$ be a non-negative random variable with twice continuously differentiable cdf $F(\theta) := \Pr(\tilde{\theta} \leq \theta)$ and $E(\tilde{\theta}) = ...
1
vote
1answer
53 views

Probability Density and Distribution of a Sphere

I am given that the density function for the radii of a sphere is constant over 0 < r<5 and zero elsewhere and am asked for calculate the density function f(r) and the cumulative distribution ...