Cdf Definition and 108 Threads

The above theorem is trying to find the pdf of a transformed random variable, it attempts to do so by "first principles", starting by using the definition of cdf, I don't understand why they have a ##f_X(x)## in the integral wouldn't ##\int_{\{x:r(x)<y\}}r(X) dx## be the correct integral for the...

The team has found that the particle, known as a W boson, is more massive than the theories predicted. The result has been described as "shocking" by Prof David Tobak, who is the project co-spokesperson. The discovery could lead to the development of a new, more complete theory of how the...

$$f_{XY}=1$$ $$dzdy=2xdxdy⇒\frac{1}{2\sqrt{z}}dzdy=dxdy$$ $$f_{ZY}=\frac{1}{2\sqrt{z}}\quad \text{on some region S}$$ $$F_{ZY}=\int^y_{g}\int^x_{h}\frac{1}{2\sqrt{z}}dzdy\quad\text{for some}\quad g(x,y),h(x,y)$$ im learning how to find the region S using a change-of variables technique

Given the probability density function f(x) = b[1-(4x/10-6/10)^2] for 1.5 < x <4. and f(x) = 0 elsewhere. 1. What is the value of b such that f(x) becomes a valid density function 2. What is the cumulative distribution function F(x) of f(x) 3. What is the Expectation of X, E[X] 4. What is...

I'm not quite sure if my problem is considered a calculus problem or a statistics problem, but I believe it to be a statistics related problem. Below is a screenshot of what I'm dealing with. For a) I expressed f(t) in terms of parameters p and u, and I got: $$f(t)=\frac{-u \cdot a + u \cdot...

Suppose I have the random variables ##Z_k=X_k/Y_k## with a PDF ##f_{Z_k}(z_k)## for ##k=1,\,2\,\ldots, K##, where ##\{X_k, Y_k\}## are i.i.d. random variables. I can find \text{Pr}\left[\sum_{i=1}^3Z_k\leq \eta\right] as...

This thread will be a collection of multiple questions I asked before over different forums. I will start from the beginning, and I hope someone will follow the steps with me, because I did it before alone, and I ended with a numerical integration that is not finite, which doesn't make sense...

Is there a way to find the CDF of a random variable from its characteristic function directly, without first finding the PDF through inverse Fourier transform, and then integrate the PDF to get the CDFÉ

Hi, I have this random variable ##\beta=\sum_{k=1}^K\alpha_k##, where ##\{\alpha_k\}_{k=1}^{K}## are i.i.d. random variables with CDF ##F_{\alpha}(\alpha)=1-\frac{1}{\alpha+1}## and PDF ##\frac{1}{(1+\alpha)^2}##. I want to find the CDF of the random variable ##\beta##. So, I used the Moment...

Homework Statement We have F(x) = ∑ (1/2)^j [Sum goes from j=1 to x] For the cdf F(x), find the pmf f(x), the 25th percentile, and the 60th percentile.Homework EquationsThe Attempt at a Solution I've been doing numerous of these for continuous distributions, however this one is tricking up...

Homework Statement Consider a continuous random variable X with the probability density function fX(x) = |x|/5 , – 1 ≤ x ≤ 3, zero elsewhere. I need to find the cumulative distribution function of X, FX (x). 2. Homework Equations The equation to find the cdf. The Attempt at a Solution FX(x)...

I am not sure where to ask this question but I must try somewhere. 1. Homework Statement I am trying to graph a simple cdf on my ti-84 and I cannot get it to work. I need an example to follow Homework Equations See below. The Attempt at a Solution 2nd, dist, 2 enter -1E99,0 0,.5 [/B]I also...

Homework Statement I was hoping someone could just verify this solution is accurate. p(x) = 0 , x < 0 4x, x < .5 -4x + 4 , .5 <= x < 1 Find CDF and Inverse of the CDF. Homework EquationsThe Attempt at a Solution CDF = 0 , x < 0 2x^2 ...

There's this problem that I've been trying to solve. I know the solution for it now but my initial attempt at a solution was wrong and I can't seem to figure out the mistake with my reasoning. I'd appreciate some help with figuring this one out. 1. Homework Statement I have a set of random...

Hello all, I have the following random variable ##X=\frac{a_1}{a_2+1}##, where ##a_i=b_i/c_i##, where ##b_i## and ##c_i## are exponential random variables with mean 1. I need to evaluate the CDF of ##X## as F_X(x)=Pr\left[X\leq x\right]=Pr\left[\frac{a_1}{a_2+1}\leq...

Hello all, I have the following random variable ##\Gamma_m=\frac{a_m}{\sum\limits_{\substack{n=1\\n\neq m}}^Ka_n+1}## where the random variables ##\{a_n\}## are independent and identically distributed random variables. The CDF of random variable ##a_n## if given by F_{a_n}(x)=1-\frac{1}{1+x}...

Hello all, Suppose I have the following summation ##X=\sum_{k=1}^KX_k## where the ##\{X_k\}## are independent and identically distributed random variables with CDF and PDF of ##F_{X_k}(x)## and ##f_{X_k}(x)##, respectively. How can I find the CDF of ##X##? Thanks in advance

Hello all, I have the random variable ##X## with CDF and PDF of ##F_X(x)## and ##f_X(x)##, respectively. Now I have a function in terms of the random variable ##X##, which is ##e^{-X}##, and I want to find the mean of this function. Basically this can be found as...

Hi If I have a CDF = integral wrt t from 0 to r^2/2 of 2*sqrt(pi*t)*e^(-t)*dt but I want to generate samples from the PDF, would Metropolis-Hastings algorithm be the best alternative? Rgds rabbed

Homework Statement Let $$ \Phi(x)=\int_{-\infty}^{x} \frac{1} { \sqrt{2\pi} } e^{-y^2 /2} dy $$ and $$ \phi(x)=\Phi^\prime(x)=\frac{1} { \sqrt{2\pi} } e^{-x^2 /2} $$ be the standard normal (zero - mean and unit variance) cummulative probability distribution function and the standard normal...

If I have the following relations: X = sqrt(1-V^2)*cos(U) Y = sqrt(1-V^2)*sin(U) Z = V where (-pi < U < pi) and (-1 < V < 1) are independent random variables, both with uniform distributions. How do I use the CDF method to find X_pdf(x)? X_pdf(x) = X_cdf'(x) = ( P( X < x ) )' = ( P(...

Homework Statement Given a Uniform Distribution (0,1) and Z = X/Y Find F(z) and f(z) Homework EquationsThe Attempt at a Solution So I'm just trying to make sure i have the range correct on this one... I'm honestly lost from beginning to end with it. R(z) = {0,∞} because as y is very small, Z...

Hi, I have trouble with the following problem: Gaussian random variable is defined as follows \phi(t) = P(G \leq t)= 1/\sqrt{2\pi} \int^{t}_{-\infty} exp(-x^2/2)dx. Calculate the expected value E(exp(G^2\lambda/2)). Hint: Because \phi is a cumulative distribution function, \phi(+\infty) =...

Problem Let X be a uniform(0,1) random variable, and let Y=e^−X. Find the CDF of Y. Find the PDF of Y. Find EY. Relevant Equations http://puu.sh/kAVJ8/0f2b1e7b22.png My attempt at a solution If I solve for the range of y I get (1, 1/e), but because Y is not an increasing function, my...

Homework Statement X ~ Uniform (0,1) Y = e-X Find FY (y) - or the CDF Find fY(y) - or the PDF Find E[Y] 2. Homework Equations E[Y] = E[e-X] = ∫0 , 1 e-xfx(x)dx FY(y) = P(Y < y) fY(y) = F'Y(y) The Attempt at a Solution FX(x) = { 0 for x<0 x for 0<x<1 1 for 1<x } fX(x) = { 1 for...

Homework Statement Suppose you take a pass-fail test repeatedly. Let Sk be the event that you are successful in your kth try, and Fk be the event that you fail the test in your kth try. On your first try, you have a 50% chance of passing the test. P(S1)=1−P(F1)=1/2. Assume that as you take the...

Homework Statement The current in a certain circuit as measured by an ammeter is a continuous random variable X with the following probability density function: f(x) = 0.057x + 0.272 if 3 <= x <= 5. It is 0 otherwise. a) Verify that the total area under the density curve is 1. b) Obtain the...

I was watching a youtube video from MIT's open courseware series on probability. A scenario was proposed: Al is waiting for a bus. The probability that the bus arrives in x minutes is described by the random variable X, which is uniformly distributed on the interval [0,10] (in minutes). I...

Homework Statement Homework Equations integral from -inf to inf of fx(x)dx = 1, fx(x)=PMF The Attempt at a Solution I get values for probability function PMF of: 0.3, x=0 0.3, x=2 A-0.6, x=3 I guess I try to find area under curve of PMF which will be equal to 1. (0.3)(2) +...

Homework Statement Let ##X## have the pdf ##f_X(x)=\dfrac{1}{\sqrt{2\pi\sigma^2}}e^{\dfrac{-(x-\mu)^2}{2\sigma^2}}## where ##-\infty<x<\infty,-\infty<\mu<\infty,\sigma>0##. Let ##Z=g(X)=\frac{X-\mu}{\sigma}##. Find the pdf and cdf of ##Z##[/B] Homework EquationsThe Attempt at a Solution...

Y and Z are independent N(0, 1) random variables. Let X = |Z|. Consider the random point (X, Y). (a) Derive the CDF FD(d) = P(D ≤ d) of the distance from the origin D = √(X2 + Y2). Sketch this CDF as a function of all real d. (b) The ratio T = Y/X has Student’s t-distribution with 1 degree...

Homework Statement If ##X## is any random variable defined on ##[0,\infty]## with continuous CDF ##F_X(t)##. Prove that ##E(X)=\int_1^\infty (1-F_X(t)) dt##. . Homework Equations The Attempt at a Solution I am not sure how to go about this. I think double integration can be used to prove it...

Homework Statement Show that the given function is a cdf (cumulative distribution function) and find F_X^{-1}(y) (c) F_X(x) = \frac {e^{x}}4 , if x<0, and 1-(\frac {e^{-x}}4) , if x \geq 0 Homework Equations for a strictly increasing cdf, F_X^{-1}(y) = x \iff F_X(x) = y and for a...

Ive attached the R ratio formula to this post, please could someone who is familiar with it tell me what Nc is? Does it relate at all to centre of mass energy? Thanks in advance for any help. source...

Hello. I was wondering if the following is correct. Let S= a*X/(b*X+c), where a,b,c are positive constants and X is a positive random variable. Also let H= h, where h is also a positive random variable (S and H are mutually independent). Then, let F_{Z}(.) and f_{Z}(.) denote the CDF...

How to transform a random variable CDF to a standard normal Given F(x) = 1- exp (-sqrt x), for x greater that 0 Thanks.

Hello, i m trying to evaluate the following: r*x - r*y ≤ g, where r,x,y are nonnegative random variables of different distribution families and g is a constant nonnegative value. Then, Pr[r*x - r*y ≤ g] = Pr[r*x ≤ g + r*y] = ∫ Fr x(g + r*y)*fr*y(y) dy, where F(.) and f(.) denote CDF and...

Homework Statement Find the CDF of f(x) = |\frac{x}{4}| if -2<x<2 \\ 0 otherwise Homework Equations The Attempt at a Solution I have to integrate the pdf and to do so, I have to split it into two parts \int_{-x}^{0}\frac{-t}{4}dt + \int_{0}{x}\frac{t}{4}dt integrating I get \frac{x^2}{8} +...

Assume X is a random variable under a probability space in which the sample space ?= {a,b,c,d,e}. Then if I am told that: X({a}) = 1 X({b}) = 2 X({c}) = 3 X({d}) = 4 X({e}) = 5 And that: P({a}) = P({c}) = P({e}) = 1/10 P({b}) = P({d}) = 7/20 Find the C.D.F of X, the density of X...

Let F be any distribution function. With either the indefinite integral, or taking limits at plus and minus infinity, is there an equivalent expression to ∫ F(x)dx ? Can we derive one? Thanks.

Please refer to the attached image. I can't quite get the bounds for question a) right. it's so confusing. Would it be wise to split the double integral into two parts? I guess that's usually favourable with when dealing with absolute values. But the bounds are still confusing me. would it be...

Homework Statement Given the function F(x) = 1 - e^(-ax) - axe^(-ax) for x>=0 and 0 elsewhere, for which values of 'a' does the function constitute a CDF? Homework Equations The Attempt at a Solution I started with saying for that to occur, provided x1>x2 -> F(x1) > F(x2)...

Please refer to the attached image. For part a) when I want to find the CDF, don't I simply take the indefinite integral of e^-|x|, multiply it by c and solve for that = 1? I am unsure of how to take the integral for this, am i correct in saying it is -e^-x, for all x ? that would leave me...

Homework Statement Homework Equations My professor gives us the following formula: [ tex ] F_j(t) = \int\limits_0^t exp\Big\{-\sum\limits_{j=1}^J \int\limits_0^u \lambda_j^{\#}(v)dv \Big\} \lambda_j^{\#}(u)du[ / tex ] where [ tex ]\lambda_j^{\#}(t) [ / tex ] are the cause-specific...

Hi, i am trying to develop a CDF from a given MGF. The standard way of using the inverse Laplace transform etc.. is not feasible due to complexity of MGF. I was woldering if there is another straighforward direction via integration or differentiation method to produce the CDF (or PDF)...

Homework Statement Let X ~ norm(5,10). Find P(X>10).Homework Equations f(x) = \frac{1}{δ\sqrt{2π}} e^{-\frac{(x-μ)^2}{2δ^2}} F(x) = P(X<x) = \int_{-∞}^x f(u) du The Attempt at a Solution P(X>10) = 1 - P(X<10) P(X<10) = \int_{-∞}^{10} \frac{1}{δ\sqrt{2π}} e^{-\frac{(x-μ)^2}{2δ^2}} dx =...

Hi there, So regular i thought that the procedure was F(s) = ∫s0 f(x) dx However i am doing a problem with a kinked pdf and it is telling me to do something like F(s) = ∫s0 f(s) ds for 0=<s>=1/2 then... F(s) = F(1/2) + ∫s1/2 f(s) ds I am confused at the process of using f(x) or...

how are cdf and pdf related in statistics? please help i have a test tomorrow

Homework Statement Let f(x) = (1 + cx)/2 for x between -1 and 1 and f(x)=0 otherwise, where c is between -1 and 1. Show that f is a density and find the corresponding cdf. Find the quartiles and the median of the distribution in terms c. Homework Equations NA The Attempt at a Solution I...

In the page that I attached, it says "...while at the continuity points x of F_x (i.e., x \not= 0), lim F_{X_n}(x) = F_X(x)." But we know that the graph of F_X(x) is a straight line y=0, with only x=0 at y=1, right? But then all the points to the right of zero should not be equal to the limit of...