# Tagged Questions

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### Lost a factor while integrating by substitution

In the article this article by Rusell May on a generalisation of the coupon collector's problem the integral (5) is simplified in a special case, resulting in the equation (7). At this point, $L$ is ...
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### Integral / Gamma Expectation

I would like to solve the following integral, $\int_{0}^{\infty}\frac{\phi}{a+b\phi} \phi^{c-1}e^{-d\phi}d\phi$. Note $\phi \sim Ga(c,d)$ is a gamma distributed random variable and the integral can ...
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### Marginal change in the expected value

Given the function $$\mathbb{E}[u(t,s,x)]=\int_{X}\max_{t}\left\{ \int_{a}^{b}u(t,s)f(s\mid x)ds\right\} f(x)dx$$ where $f()$ is the pdf, and $X$ is the sample space of $x$. How can i analyze ...
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### moment generating function of a shot noise process

A general setting shot noise process $X(\tau)=\sum \limits_k Z(\tau, T_k)$ where $T_k$ Poisson process with intensity $\lambda(t)$, $Z(.,t)$ independent stochastic processes. Show that the moment ...
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Suppose $X$ is distributed exponentially with parameter $\lambda$. Its pdf is $\lambda e^{-\lambda x}$, and the calculation of its expectation is straight forward: $\mathbb{E}(X) = \int_0^\infty ... 1answer 36 views ### derivative of expected value of maximum of two stochastics variables (iid) I need to optimize an expected value of a maximum value for$q$. The problem has three variables,$q$is a constant and$D_1$and$D_2$are stochastic variables with pdf$f(x)$and cdf$F(x)$. The ... 0answers 40 views ### Reference for theorem? Inequality of integrals of increasing function over two distributions I have a monotone increasing function$H(x)$and two distributions with CDFs$F_1$and$F_2$, where$F_1(x) \leq F_2(x)$everywhere. The domain is$[0,\infty)$. This seems like it must be true: $$... 0answers 27 views ### Probability density function expected values given a cumulative distribution function I have this probability problem that reads: suppose f(x) is the probability density function of X where f(x)=0 unless the values of x either are or in between b and zero (0 \leq x \leq ... 1answer 33 views ### Finding conditionally expected y given a specific x from a joint distribution function! I want to determine expected y, given x=2 given joint pdf shown below$$\frac{1}{2y} * e^{-\frac{y^2 + \frac{x}{2}}{y}}$$for x,y \gt 0 and 0 otherwise I believe this means I want ... 0answers 13 views ### Implication of Monotone Likelihood Ratio Property I can't figure out how a paper is using the monotone likelihood ratio property (MLRP). Here is the statement of the MLRP: The conditional distribution G(\sigma|c) has positive density ... 0answers 30 views ### Choosing the integral limits for marginal distribution I've been trying to understand the following: The distribution of two continuous random variables is given by$$f_{X,Y}(x,y)=\frac{3}{7}x\space\space 1\le x\le 2,0\le y\le x$$and 0 otherwise. ... 0answers 33 views ### Simplification of Double Integral with Independent Parameters I am trying to find a posterior distribution and the hint is that the double integral in the denominator should simplify because p1 and p2 are independent. \displaystyle \int$$\displaystyle ... 1answer 45 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 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: $$f(x) = ... 0answers 41 views ### the marginal pdf of a transformed variable from a joint distrubution The questions tells us to let X and Y be random variables for which the joint p.d.f. is as follows:$$f(x,y)= \begin{cases} 2(x+y), & \text{for$0 \le\ y \le\ x \le\ 1$} \\ 0, & ... 1answer 56 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? 0answers 49 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 ...
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 ...