Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

Questions tagged [probability-calculus]

20 questions
14
votes
7 answers

Intuitively understand why the Poisson distribution is the limiting case of the binomial distribution

In "Data Analysis" by D. S. Sivia, there is a derivation of the Poisson distribution, from the binomial distribution. They argue that the Poisson distribution is the limiting case of the binomial distribution when $M\rightarrow\infty$, where $M$ is…
Ytsen de Boer
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7
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1 answer

Reason for absolute value of Jacobian determinant in change-of-variable formula?

When we have a random variable $x$ with a probability density $p(x)$, and a function $y = f(x)$ that is differentiable and can be solved for $x = g(y)$, the change of variable formula leads us to a density for $y$ given by $$ p(x) \, dx = p(x)…
Durden
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5
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1 answer

Do the parameters that arise in de Finetti`s representation theorem follow the rules of probability?

I recently stumbled upon de Finetti`s (pretty cool) representation theorem (What is so cool about de Finetti's representation theorem?). I wondered whether the RV $\Theta$ that arises in this context follows the rules of the probability calculus. Or…
Sebastian
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5
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2 answers

Expected value of a function including the cumulative normal distribution

Consider the function $Y = 1 - 2 \Phi((c_j - \mu)/\sigma) + 2 \Phi^2((c_j - \mu)/\sigma)$ where $\Phi$ is the cumulative distribution function for the standard normal distribution and $c_j$ is a uniformly distributed random variable on the range -1…
user2728808
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4
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2 answers

Using Chebyshev's inequality to obtain lower bounds

Let $X_1$ and $X_2$ be i.i.d. continuous random variables with pdf $f(x) = 6x(1-x), 0
Shreya Bhandari
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4
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1 answer

support of an importance sampling with respect to the original distribution function

I have a question regarding the support of an importance sampling distribution with respect to the support of the original distribution function. I was reading that the support of the importance sampling distribution (say the importance distribution…
john_w
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3
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1 answer

Integral formula using R

I have to implement this formula: $K(x) = \int_{0}^{0.5}q_{\theta}(x)d{\theta}$ where $q_{\theta}(x)$´s are the conditional quantiles in some $\theta$. using a range of $\theta = [0.45; 0.40; 0.35; 0.3; 0.25; 0.2; 0.15; 0.1; 0.05]$ Note that i am…
Linkman
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1 answer

Derivative of expectation where the variable appears in the integration limit and in the integrand?

I want to calculate the derivative of $$\varphi(\mu) = \int_{-\infty}^{\mu} r(x-\mu) f(x)dx,$$ wrt to $\mu$, where $r$ is a function and $f$ is a density function. How can I account for the presence of $\mu$ in the integration limit and in the…
Martell
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1 answer

$k$-th order statistics when the value of $j$-th one is known

Suppose there are $n$ random variables $X_i,~i\in\{1,\cdots,n\}$ which are independently drawn according to a CDF $F$ and pdf $f$. Suppose also that we know one of the realization, say $X_{(j)}=x_{(j)}$, and we also know that it is the $j$-th…
Andeanlll
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1 answer

Prove that E[x^n] >= (EX)^n for n = 2k

Prove that $E[x^n] \geq (Ex)^n$ for $n = 2k$ I only have the formula of E(x) but I don't know how to prove it.
Ann Height
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vote
0 answers

Solve probability equation for one variable

Given: p = $a^l$ * ($\frac{c}{a^l + b^l}$ + $\frac{b}{a^l + c^l}$) Where a, b, c, p are known and are probabilities. Solve for l. (1 equation and 1 unknown) Does a closed form solution to this exist? I can't see how to solve using algebra. If there…
Savino
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1 answer

Probability mass function from multiple datasets with different ranges

Assume 3 datasets (experiments) for the same population of a discrete random variable, dataset 1 has observed values {1, 2, 3, NA} values, dataset 2 {1, 2, 3, 4, 5, NA} and dataset 3 {1, 2, 3, 4, 5, 6, 7, NA}. These are very large datasets (in the…
user90772
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1 answer

Interchanging limit and derivative for CDFs

Let $F_{\theta}(x)$ denote a cumulative distribution function indexed by the parameter vector $\theta$. Given this definition is the following equation correct (and if so under which conditions)? $$\lim_{\theta \rightarrow…
user304347
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3 answers

How a statistical package like SAS analyses market risk without any calculus support

SAS is a very popular tool to analyze the market risks of a portfolio of stocks,bonds including non linear components like options. Assuming that option analysis uses advanced calculus and even stock price modelling uses stochastic differential…
Victor
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What does correlation mean in error propagation?

From the python uncertainties package: Correlations between expressions are correctly taken into account. Thus, x-x is exactly zero, for instance (most implementations found on the web yield a non-zero uncertainty for x-x, which is incorrect). x…
naught101
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