# Questions tagged [cumulative-distribution-functions]

For questions related to cumulative distribution functions.

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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $$F_X(x) = \... 1answer 47 views ### Find the CDF of Y=X+|X-a| where X\sim\text{unif}[0,b], b>a>0 Given X\sim\text{unif}[0,b], I need to find the following probability:$$F(y)\triangleq\mathbb{P}(Y\leq y)$$For all y\in\mathbb{R}, where Y=X+|X-a| and b>a>0 are given positive ... 2answers 62 views ### How to test if numerical function describes a valid probability distribution? Suppose I can query a function f, but I don't have its closed form. We know the following things about f: f(x) \geq 0 for all x f is continuous Additionally, I can choose whether f(x) \leq ... 1answer 52 views ### why is F_Y (y) = F_X(x)? [closed] Assuming y=g(X), why is F_Y(y)=F_X(x)? I know that if g(x) is a strictly monotonically increasing function of x the above holds true, but I do not know how to explain it. edit: i am told that ... 1answer 34 views ### CDF in Probability$$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.$$Given that P(-3 \leq x < 3) = \frac{... 1answer 63 views ### Bayes' theorem and law of total probability with CDFs Suppose X has Gamma(2, λ) distribution, and the conditional distribution of Y given X = x is uniform on (0, x). Find the joint density function of X and Y, the marginal density function of ... 1answer 46 views ### Inverse of cumulative distribution function Let F(x) is the cumulative distribution function and P(x) is the (given) probability distribution function and X is a random variable. Can anybody please intuitively explain, Why can the ... 0answers 26 views ### Finding expectation of continuous random variable [duplicate] Let X be a continuous random variable with CDF F. Suppose that P(X>0)=1 and that E(X) < \infty. Show that E(X) = \int_{0}^{\infty}P(X>x)dx I start out with the definition of ... 0answers 28 views ### Inverse distribution function and random walk Suppose P denotes the probability distribution function of a random walk \sum^n X_i and is given already. Can anybody please intuitively and mathematically explain, how the inverse distribution ... 0answers 56 views ### What is the relationship between two formulas for expected value in terms of CDF? There are two formulas for expected value in terms of CDF:$$ 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 $$See e.g. Wikipedia. Are they ... 1answer 51 views ### How to find the density function of |X|^{\frac{1}{2}} where X follows a standard normal distribution? I have to find the density function of |X|^{\frac{1}{2}} where X follows a standard normal distribution. This is what I have attempted so far:$$F_y(y)=P(Y\le y)=P(|X|^{\frac{1}{2}}\le y)=P(|X|\le ...
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 ...