Questions tagged [cumulative-distribution-functions]

For questions related to cumulative distribution functions.

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8 views

Standard normal cumulative distribution function

I am working on a problem where I want to find the distribution of $X$. $$P(X \le x) = P(Z \le x, U \le \frac{1}{2}) + P(-Z \le x, U \gt \frac{1}{2})$$ Where $Z$~$N(0,1)$, $U$~$(0,1)$ and $Z$ and $U$ ...
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11 views

Negatives and cumulative distribution functions

I am working on a problem where I am dealing with $$P(-Z \le x)$$ where Z is a standard normal random variable. I am trying to figure out how to get to a cdf from here but I am not sure if I am using ...
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23 views

Calculating the probability that X = Y from the CDF alone

Give two random variables X and Y with joint CDF $F_{X,Y}$, I am interested in calculating $P(X=Y)$. I do not assume $F_{X,Y}$ is absolutely continuous. Is it possible to calculate this probability ...
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22 views

Finding distribution of X

We have X=Z if U<=0.5 and X=-Z if U>0.5 where Z is a standard normal variable and U is a uniform random variable (0,1). I want to find the distribution of X. I am mainly unsure of how to ...
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9 views

If $F$ satisfies conditions for a c.d.f., then it is a c.d.f. for some random variable?

Larry Wasserman , in his All of Statistics, states that for any cdf $F$ : $F$ is non-decreasing $\lim_{x\rightarrow -\infty} F = 0 \ \ \text{and} \ \ \lim_{x\rightarrow \infty} F = 1$ $F$ is right ...
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1answer
18 views

Finding the form of the equation of the curve given some conditions

I am having difficulties to find the form of the equation of the curve $y=f(x)$. $f(x)$ has the all of the following properties: it is continuous (and not piecewise), always increasing, having at ...
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1answer
23 views

How to calculate CDF when X is discontinued?

recently I am doing a question I want to find P(2 < x ≤ 3) I can tell P(x ≤ 3) is 0.4 [P(5 ≤ x) + P(3 ≤ x < 5) = 0.2 +0.2)] However, how can I get 2 < x? I want to find P( x = 3) However,...
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17 views

Properties of CDF Beta distribution

Let $F^{Beta}_{\alpha, \beta}(x)$ denote the CDF of beta distribution with shape parameters $\alpha$ and $\beta$. Let $\rho \in (0,1)$. Suppose $\ell_1 < \ell_2$. Is the following inequality true ...
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1answer
23 views

Given $F$ is a CDF, what can we say about $H(x) = 1 - (1-F(x))^2$?

Let's say we know that a function $F$ is a CDF. What about the function $H(x) = 1 - (1-F(x))^2$? I know there's a theorem that says a function is a CDF if $\lim_{x \to -\infty}F(x) = 0$ and $\lim_{x \...
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1answer
19 views

Joint density of two perfectly correlated RVs

Consider two RVs $X_1$, $X_2$, where the density of $X_1$ is $p_{X_1}(\cdot)$ while $X_2 = X_1-x_0$ for some costant $x_0$, i.e. $X_2$ is a simple translation of $X_1$. I want to find, if it possible, ...
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12 views

Deducing the Cumulative Function of the Cauchy Distribuition

The Cauchy Distribuition: $$f(x;x_{0},\gamma )={\frac {1}{\pi \gamma \left[1+\left({\frac {x-x_{0}}{\gamma }}\right)^{2}\right]}}={1 \over \pi \gamma }\left[{\gamma ^{2} \over (x-x_{0})^{2}+\gamma ^{...
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1answer
30 views

Does $Pr(F_X(x) < t) = Pr(x < F^{-1}(t))$?

Consider a random variable $X$ with CDF $F_X(x) = Pr(X \leq x)$. I am wondering if $Pr(F_X(x) < t) = Pr(x < F^{-1}(t))$ holds in general for any $t$?
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58 views

Some properties of Owen's T function

Owen's $T$ function is given by $$ T(h,a) = \frac{1}{2 \pi} \int_0^{a} \frac{\mathrm{e}^{-\frac{h^2}{2} (1+x^2)}}{1 + x^2} \mathrm{d}x $$ There are some properties of the function, which one can find ...
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1answer
25 views

Cumulative Distribution Function of $X+Y$, where $X,Y$ are independent is convolution of $F_X$ and $F_Y$?

I'm reading Introduction to Probability Models by Sheldon Ross, 12th edition. On page 57, it says: Suppose first that $X$ and $Y$ are continuous, $X$ having probability density $f$ and $Y$ having ...
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1answer
25 views

Evaluate g'(x) in terms of $\phi$ where $\phi$ is the CDF of the $N(0,1)$.

For the following expression could someone provide some help/suggestions of my progress please $g(x) = \int_{-ln(x)}^{x^2} \phi(t+x) dt $ The question asks me to evaluate $g'(1)$ in terms of $\...
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1answer
65 views

A Seemingly Discrete Distribution Which Is Actually Continuous.

I have been working on my Statistics recently, and while working through the book I have found a problem I didn't quite understand. The setting is like this: A box contains good/defective items mixed ...
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15 views

How to get cumulative probability from truncated Gaussian distribution?

I have this truncated normal pdf: $$ p ( x ) = \frac 1 Z \exp \left( - \frac { ( x - \mu ) ^ 2 } { 2 \sigma ^ 2 } \right) $$ where $ Z = \sqrt { \frac \pi 2 } \sigma \left[ \operatorname {erf} \left( \...
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29 views

expected value of non-negative random variables

Let $P(Y \ge y) = {1 \over y \log(y)}$ and $y \ge e$, then $F(y) = 1- {1 \over y \log(y)}$ and $F' = {\log(y) + 1 \over (y \log (y))^2}$. Assuming that $Y$ is non-negative, you can get the expected ...
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25 views

Upper bound on the CDF of the sum of positively correlated random variables with fixed marginals

Let $X_1$ and $X_2$ be (positive and) positively correlated random variables on $\mathbb{R}$ ( $\mathbb{E}(X_1 X_2) - \mathbb{E}(X_1)\mathbb{E}(X_2) \ge 0$) with equal marginal distribution ( Law($X_1$...
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21 views

Central limit theorem for proving equation

Suppose $X_1, X_2, ..., X_n$ are sequences of iid random variables with mean equal to zero and variance = $σ^ 2$. we define $Y_n = \frac{s_n}{σ∗\sqrt{n}} −\frac{s_{2n}}{σ∗\sqrt{2n}}$ , $S_n = X_1 + ...
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35 views

Finding PDF of a random variable From its CDF

Let X be random variable with cumulative distribution function $P(X\leq x) = \left\{ \begin{array}{ll} 0, \quad \quad x<0,\\ \frac{x}{8}, \quad \quad 0\le x < 2 \\ \frac{x^2}{16},\quad \quad 2\...
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38 views

Are there “continuous” random variables without a density function?

Assume you have a random varaible with a cumulation distribution function $F(x)=P(X\le x)$. This function will always be right-continuous, but assume that it also is left-continuous. Then there are no ...
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14 views

What does ratio of probability density of two points signify?

It's known that in a continuous distribution, pdf value for a given point has no significance. The absolute likelihood for a continuous random variable to take on any particular value is 0 (since ...
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27 views

Understanding a Probability Distribution with a CDF and how to use the CDC's asymptomatic estimate thoroughly

I'm having a hard time understanding how an academic paper link derived their asymptomatic case count. The equation they used was: $$ g(x,p)=\left\{ \begin{array}{ll} ...
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34 views

Expectation of a product between a random variable and the logistic function

Consider a Gaussian variable $x$ with mean $\mu$ and variance $\sigma^2$, can I calculate or approximate $\mathbb{E}\!\left[x\, \phi(x)\right]$, such that $\phi$ is the sigmoid function given by $\phi(...
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2answers
26 views

What does the subscript of a CDF mean?

I am working with the composition method for generating a random variable $X$. I have always seen CDFs denoted as $F_X(x)$ but my question is what does it mean if the CDF is $F_I(X)$. So specifically ...
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2answers
57 views

Asymptotic behavior of the CDF of a $\operatorname{Beta}(n,n)$ at $x<1/2$.

I know that the if $X_n \sim \operatorname{Beta}(n,n)$ (see here for a definition) then $\mathbb{P}_{X_n} \to \delta_{1/2}, n \to \infty$ weakly. If $\varepsilon \in (0,1/2)$, I'm wondering about how ...
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1answer
20 views

Derivative of integral involving differential of CDF

I am struggling to compute the following partial derivative of an integral $\dfrac{\partial }{\partial t} \int_{a}^{\infty} x(t) dF(x(t)) $, where x is a random variable that depends on the ...
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11 views

How to fit CDF of normal distribution using maximum likelihood?

I have data given as counts for each threshold value (so I have thresholds $[x_1, x_2...,c_L]$ and corresponding counts $[c_1, c_2,...,c_L]$ with $N$ trials each), which would correspond to observed ...
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1answer
18 views

Is there a discontinuous random vector with continuous components?

Is it possible that $X$ and $Y$ are continuous random variables (i.e., their cdfs are continuous), yet the random vector $(X,Y)$ is discontinuous (i.e., their joint cdf is discontinuous) ?
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14 views

Finding the pdf of another RV using a given joint pdf

Question: Let $S$ and $T$ be jointly continuous random variables with joint pdf $f(s,t)$. Find an expression for the density of $W = S - T$ in terms of the joint pdf $f$. My work so far: ...
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1answer
36 views

Geometric distribution where failure probability is not 1-p

The typical geometric distribution is defined from the success probability $p$, i.e., a r.v. G~Geometric($p$), would have PMF... $$P[G=g]=(1-p)^{g-1}p$$ I have this problem from Bertsekas: For part a,...
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1answer
34 views

Different definitions of independent random variables

Let $X$ and $Y$ be random variables with a joint density function. In some books, the independence of $X$ and $Y$ is defined as \begin{equation} P(X\in A,\ Y\in B)=P(X\in A)P(Y\in B) \tag{1} \end{...
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1answer
30 views

Moments Dominance and First Order Stochastic Dominance

If two random variable satisfies $EX^n \leq EY^n$ for all $n=1,2,3,...$, can we say Y First order stochastically dominates X? i.e. $P(X<t)>P(Y<t)$ for all t I have been thinking since we can ...
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21 views

Determining the areas above, below, and between two cumulative distributions

I have two cumulative distributions, which reflect two different beta distributions (one with mean = $.5$, precision/phi =$ 3$, the other mean is $.35$, and the precision/phi is $20$). The cumulative ...
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62 views

Derive variance of log-normal distribution from mean and cumulative distribution function

How can the variance $\sigma_x^2$ of a log-normal distribution be derived from its mean value $\mu_x$ and a target value $\hat{x}$ for a fixed value of its cumulative distribution function $$F_X(x) = \...
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1answer
47 views

Find the CDF of $Y=X+|X-a|$ where $X\sim\text{unif}[0,b], b>a>0$

Given $X\sim\text{unif}[0,b]$, I need to find the following probability: $$F(y)\triangleq\mathbb{P}(Y\leq y)$$ For all $y\in\mathbb{R}$, where $Y=X+|X-a|$ and $b>a>0$ are given positive ...
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2answers
62 views

How to test if numerical function describes a valid probability distribution?

Suppose I can query a function $f$, but I don't have its closed form. We know the following things about $f$: $f(x) \geq 0$ for all $x$ $f$ is continuous Additionally, I can choose whether $f(x) \leq ...
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1answer
52 views

why is $F_Y (y) = F_X(x)$? [closed]

Assuming $y=g(X)$, why is $F_Y(y)=F_X(x)$? I know that if $g(x)$ is a strictly monotonically increasing function of $x$ the above holds true, but I do not know how to explain it. edit: i am told that ...
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1answer
34 views

CDF in Probability

$$f(x)=\left\{\begin{array}{ll} C & \text { for }-3 \leq x<3 \\ Dx & \text { for } 3 \leq x<5 \\ 0 & \text { otherwise } \end{array}\right.$$ Given that $P(-3 \leq x < 3) = \frac{...
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1answer
63 views

Bayes' theorem and law of total probability with CDFs

Suppose $X$ has Gamma(2, λ) distribution, and the conditional distribution of $Y$ given $X = x$ is uniform on $(0, x).$ Find the joint density function of $X$ and $Y,$ the marginal density function of ...
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1answer
46 views

Inverse of cumulative distribution function

Let $F(x)$ is the cumulative distribution function and $P(x)$ is the (given) probability distribution function and $X$ is a random variable. Can anybody please intuitively explain, Why can the ...
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0answers
26 views

Finding expectation of continuous random variable [duplicate]

Let $X$ be a continuous random variable with CDF $F$. Suppose that $P(X>0)=1$ and that $E(X) < \infty$. Show that $E(X) = \int_{0}^{\infty}P(X>x)dx$ I start out with the definition of ...
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28 views

Inverse distribution function and random walk

Suppose $P$ denotes the probability distribution function of a random walk $\sum^n X_i$ and is given already. Can anybody please intuitively and mathematically explain, how the inverse distribution ...
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56 views

What is the relationship between two formulas for expected value in terms of CDF?

There are two formulas for expected value in terms of CDF: $$ E(X)=\int_{-\infty}^{\infty}xdF_X(x) $$ $$ E(X)=\int_{0}^{\infty}(1-F_X(x))dx-\int_{-\infty}^{0}F_X(x)dx $$ See e.g. Wikipedia. Are they ...
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1answer
51 views

How to find the density function of $|X|^{\frac{1}{2}}$ where $X$ follows a standard normal distribution?

I have to find the density function of $|X|^{\frac{1}{2}}$ where $X$ follows a standard normal distribution. This is what I have attempted so far: $$F_y(y)=P(Y\le y)=P(|X|^{\frac{1}{2}}\le y)=P(|X|\le ...
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1answer
73 views

A version of fundamental theorem of calculus

Let $\mu$ be a finite measure defined on the Borel subsets of $\mathbb{R}$ such that $\forall t \in \mathbb{R}, \mu\big(\{t\}\big)=0$; $F: \mathbb{R} \to \mathbb{R}$ be a continuously differentiable ...
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1answer
40 views

Formula for the multivariate cumulative distribution function (continuous case)

Consider a random vector of dimension $5\times 1$, $X\equiv (X_1, X_2, X_3, X_4, X_5)$, with joint CDF denoted by $G$. Consider the following probability $$ p\equiv Pr(X_1\geq a_1,X_2\geq a_2,X_3\geq ...
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2answers
40 views

CDF approximation - L'Hopital's rule

The following CDF, \begin{equation} F_{y}(x) = 1- \Big( \frac{ (1-\phi) x}{\phi (k-1)}+1\Big)^ {1-k} e^{- \frac{x}{\phi y}} \end{equation} is approximated for $k \rightarrow \infty$ as follows \...
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2answers
70 views

Questions about definition of Quantile function

Let $F$ be a distribution function. For $0<p<1$, the $p$-th quantile or fractile of $F$ is defined by $$\xi_p = F^{\leftarrow}(p) = \inf\{x:F(x)\geq p\}$$ my questions are following: Why we ...

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