Questions tagged [cdf]

Cumulative distribution function. While the PDF gives the probability density of each value of a random variable, the CDF (often denoted $F(x)$) gives the probability that the random variable will be less than or equal to a specified value.

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62 views

How do I get the CDF of a gamma distribution with mean and sd?

I have the mean and standard deviation of my data, which I determined follows a gamma distribution. I don't understand the function I found online for the CDF of a gamma distribution because of the ...
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31 views

Analytical solution to the multivariate CDF given multivariate pdf

Is there any way of approximating or analytically solving the below CDF (let's say even for $n\to\infty$)? I am trying to find the below probability: \begin{align} &P\left[X_{2}-X_{1} \leq 0,...
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How to prove KL(q(z)|p(z)) = E_q(z) [ log f(z) ] where f is the CDF of p?

As title. It was used in https://arxiv.org/abs/1905.10549 without proving.
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If $X$ follows standard normal distribution, find the correlation coefficient between $X$ and $\Phi(X)$

If $X$ follows standard normal distribution, find the correlation coefficient between $X$ and $\Phi(X)$, where $\Phi(X)$ is the cdf of $X$. My attempt is: First we have to calculate $Cov(X, \Phi(X))$...
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Derivation of CDF of a function that results in an exponential distribution

I was looking through wiki's treatment on the title topic in https://en.wikipedia.org/wiki/Random_variable and am completely stumped on this particular section: There are several specifics that ...
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Hypothesis test of distributions from two biased samples using NPMLEs

Suppose $X_1, ..., X_n$ is a biased sample (bias mechanism known) from distribution function $F_1$ and suppose $Y_1, ..., Y_m$ is another set of observations sampled in a differently known biased ...
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Can I increase the sample size by generating random numbers to apply the Chi-Square Goodness of Fit Test?

Does increasing the sample size by random number generation change the distribution? I have a sample of size 8. Each sample value represents the number of bus arrivals at a bus stop every 15 minutes. ...
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How to find quantiles and likelihoods of mixture distributions?

My PDF: M was estimated and found to be 5. I need to work out the quartiles for the PDF above. In addition, I need to use different methods of estimation to estimate the parameters. So far I've ...
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Tcdf and invT with incomplete beta [duplicate]

I am working in python. I have a function for incomplete beta. Iow would I calculate tCDF and invT using this? the incomplete beta function that I have is this: ...
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Joint CDF of M(t) and B(t), where B(t) is the standard BM and M(t) is maximum value of standard BM on [0,t]

We have to find - $F_{M(t),B(t)}(m,x) = P(M(t) \leq m, B(t) \leq x)$. $T_{m} = inf\{t\geq 0: B(t) = m\}$. We know that, $$ P(M(t) \geq m, B(t) \leq x) = P(T_{m} \leq t, B(t) \leq x)$$, $$ = P(T_{m}...
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Strategies for predicting values of a time-dependent CDF, given covariates

I've got data that looks like this: ...
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Consistent estimator and distribution function

a general question: If the distribution function $F_n$ of some estimator $T_n$ suffices \lim_{n \rightarrow \infty} F_n(x) = 1 \text{ or } 0 \forall x}. Does that imply that $T_n$ is consistent? I ...
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Calculate the derivative of the CDF with respect to the mean value [duplicate]

I want to derive the cumulative density function (cdf) for variables following normal distribution with respect to the parameters of the cdf (such as the mean or the standard deviation)
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Conditional transformation of variables

I've seen a trick for finding the p.d.f of $r(X,Y)$ where $X$ and $Y$ are r.v's by first calculating the cdf i.e $P(r(X,Y) \leq l)$ and then differentiating to find the pdf. So if $\Omega = \{(x,y) | ...
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Multiply CDF by constant, what is the expected value of this “new” CDF? [closed]

Specifically, I want to multiply $F_X(x)$ by $E(X)$, so I have $$ ??? = E(X)\cdot F_X(x) = \int^b_a xf_X(x)dx\cdot \int^b_a f_X(x)dx \overset{?}{=}\int^b_ax\Big(f_X(x)\Big)^2dx $$ Is there a way to ...
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What is the expected value of half a standard normal distribution?

You have a normal distribution with mean of 0 and variance of 1. Keeping the same probabilities and focusing only on half of the distribution (other half has it's original probabilities but x values ...
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can you help me to do a difference of CDF?

I have 2 CDF's with equal number of points that I want to compare. These are from: Temperature of 1 month from 2012 Mean temperature across months What can I do to obtain this difference, this is ...
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cumulative distribution functions (CDFs) [duplicate]

i want to know why is important use CDF for this analysis of TMY (typical meteorological year) because i have the data of the month for compare with the long term mean This the example of the manual ...
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How to find the conditional CDF based on observed data in R [closed]

If we have two samples (generally their distribution is not known),say $X\sim N(0,1)$, $Y|X\sim N(X,X^2/2)$. Can we recover the conditional CDF of $Y|X$ based on the observed samples in R? ...
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Approximate density from moments and quantiles, then sample from it

Situation I need to send R code to a third party to run estimations for me (I will not be able to work with the data directly). I want to simulate data to test some of the estimators before sending ...
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If $F_X(z) > F_Y (z)$ for all $z\in \mathbb{R}$ then $P(X < Y ) > 0$?

I came across this question in a review of an old exam I took. I didn't get the answer correctly then, and I'm struggling to figure the answer out now. Can anyone help me reason through this? ...
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Integration by parts for multivariable cumulative distribution function

How can I integrate by parts $$\int_A (y_1+\cdots+y_n) \,dF(y_1,...,y_n),$$ where $F$ is the cumulative distribution function for some random vector, $A$ is some Borel bounded set in ${\mathbb R}^n$.? ...
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Upper bound of normal cdf

Random variable $X\sim N(0,1)$. Show that, $P(X\geq c) \leq e^{-ct+ \frac{t^{2}}{2}}$ for $c>0$ and for all $t$ in $R$. I found that $P(X\geq c) = \Phi(-c)$ where $\Phi(x)=\int_{-\infty}^{x}\phi(u)...
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Mapping a range of values such that the resulting distribution is uniform [duplicate]

I have a set of values. Let's call the set X with values ... . Those values in [0, 1] have a non uniform distribution (empirically measured). I would like to re-map those values on [0, 1] such that ...
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An approximation to the cdf of the normal from a pdf?

In this paper (p. 36), authors wrote $$p(n,T) = \Phi \Big(\frac{n}{T},\mu,\sigma \Big) - \Phi \Big (\frac{n-1}{T},\mu,\sigma \Big)\; (3) $$ Bellow we will use the approximation $$p(n,T) =...
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Simulation: Generate random numbers that cluster around an average? [closed]

I want to simulate a simple event that has variable empirical result/outcome. Generate random numbers that cluster around an average For example, let's say we collect the data for how far people can ...
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Kolmogorov–Smirnov test on text data

The Kolmogorov–Smirnov test a very efficient way to determine if two samples are significantly different from each other or whether the CDF between two different samples fit each other. This can be ...
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1answer
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Efficiently Computing The Beta CDF [duplicate]

I am using numba to JIT compile some looped python functions as part of a larger application. Ideally, everything will run in numba's "no python" mode, such that the loop can be parallelised. One of ...
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Finding the joint CDF using the joint PDF; why can't I do this?

Find the joint CDF of the independent random variables $X$ and $Y$, where $f_x(x)=x/2, 0\le x \le 2, $ and $f_Y(y)=2y, 0 \le y \le 1$. To do this, we can find the CDF separately for each of the ...
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Compute $P(Y<3X)$ using joint PDF

I'm given a joint pdf $f_{X,Y}(x,y)=2e^{-x-y}, 0<x<y, 0<y $ and asked to compute $P(Y<3X)$. To do this, I let $Y=3X$ (the boundary) and found that the region of integration is under this ...
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Tail behaviour of normal cdf?

Q: What is the tail behaviour of $\log \Phi(t)$ as $t \to \infty$? Since $\Phi(t) \to 1$ as $t \to \infty$, we know that $\log \Phi(t)\to 0$, but I would like to know at what rate this function ...
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Order Statistics; Finding the probability that the first sample is < 0.6, and the last sample is > 0.6

Here is the problem statement below: A random sample of size 5 is drawn from the pdf $f_Y(y)=2y, 0\le y \le1$. Calculate $P(Y_1^{'} < 0.6 < Y_5^{'})$. Here, using formulas for order ...
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Definition of CDF of discrete RV

In many different (serious and good) statistics books I find different definitions of CDF of a discrete RV. The difference is the equal sign at the index of the summation sign. The first is: $$F(x) = ...
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Convergence in Distribution, Argument Converging in Probability

Suppose $\lim_{n\to\infty}P(X_{n}\leq x) = P(X\leq x)$ and that $A_{n} \stackrel{p}{\longrightarrow} a$, where $a$ is a continuity point of $F_{X}(x) = P(X\leq x)$. Is it the case that $\lim_{n\to\...
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Finding the CDF given marginal PDF's; setting bounds

In this question, I'm having a hard time understanding how specifically to set the bounds for the CDF. Let $X$ and $Y$ be independent variables. Find the CDF of $W=Y/X$ using the marginal PDFs ...
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PDF transformation for y=|x|

Suppose I have the random variable X with a pdf: $$f(x)=exp(-(x+1)) u(x+1)$$ where u is the unit step function; such that u = 0 for x<-1 and u=1 for x>-1 $$y= |x|$$ for $$-1<x<1$$ ...
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How to relate beta CDF to student-t CDF? [duplicate]

We can relate the student-t and beta distributions as such: If $X$ has a Student's t-distribution with degree of freedom $\nu$ then one can obtain a Beta distribution: $$\frac{\nu}{\nu + X^2} \...
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How to approximate the student-t CDF at a point without the hypergeometric function?

Is there a way to closely approximate the CDF of a student-t distribution at a point $x$ without involving the hypergeometric function? For example, by using a series expansion, or expressing the CDF ...
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1answer
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CDF Variable Transformation

Let $X$ be uniform on $(-1, 2)$ and let $Y = X^2$. Find the pdf of $Y$. So far I have noted that $F_X(x) = P(X \leq x) = \int_{-1}^x \frac{1}{3} dt = \frac{1}{3}(x+1)$. Then, since $Y=X^2$, $y \in [...
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When is the pmf of the difference of two independent random variables symmetric in zero?

Consider the stepwise cumulative distribution function $$ \Delta(x; \lambda, \mu)=\sum_{j=1}^J \lambda_j 1\{x\geq \mu_j\} \hspace{1cm} \forall x \in \mathbb{R} $$ where $J<\infty$ $\lambda\equiv (\...
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cumulative distribution function for non-normal distribution

From this article, I read that the author drew four versions of CDFs each plotted in different distributions (all four plots come from the same sample data) From these four plots, the author chooses ...
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Must the domain of a CDF be $\mathbb{R}$ or can it also be a strict subset?

So my question is whether the domain of a cumulative distribution function has to be $\mathbb{R}$ or whether it can also be a strict subset. The reason I'm asking is because I'm currently going ...
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Calculating a Confidence Interval for a Proportion for a Sample of Different Size

I'm interested in a (preferably analytic) solution or approximation to the following problem: Let $s_1$ be a sample from an unknown distribution of size $N_1$ and with proportion of successes $p_1$. ...
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How to see this order statistic result and find my error

Let W be a random variable with pdf $f(w)=\theta B^{-\theta}w^{\theta-1}$ for $0 \lt w \lt B$ and 0 otherwise. Assuming Independence, Show that , $W_{n:n} \to B$ as $n \to \infty$ where $W_{n:n}$...
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1answer
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What does area under this density plot gives me?

I am new to data science and trying to grasp the concepts. I have a question in my exercise asking "what proportion of US states have populations larger than 10 million?". The density plot is shown ...
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Under what conditions does the two-sided DKW inequality become a strict equality?

If the two-sided DKW inequality is tight, then there should be a choice of distribution and sample size where the equality holds. What is it?
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What is the problem in my CDF derivation?

Let $Z = \frac{XY}{aX+bY+c}$ where the random variable $X$ and $Y$ follows gamma distribution such that $X\sim G(\lambda_x,\theta_x)$ and $Y\sim G(\lambda_y,\theta_y)$ The CDF of $Z$ can be ...
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CDF at n of normal distribution to the nth power

I'm working with an equation that includes a normally distributed investment return R. I can find the Cumulative Distribution Function of R for the first period n=0. However, how do I derive the CDF ...
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1answer
36 views

Probability of Ranking

Suppose that I have 3 normal distributions (I wish to extend this to K), such that $p_k\sim\mathcal{N}(\mu_k,\sigma_k^2)$. Assume that $(\mu_k,\sigma_k^2)$ are known. How would you go about ...
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2answers
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Normal Quantile Function With a lower bound not equal to infinity

I was recently at a statistics competition and a question came up as follows: They drew a normal distribution with $\mu=7$ and the area between the values $7.75$ and $8.25$ equal to $0.12$. No other ...