Questions tagged [change-of-variable]

This concern all problem requesting techniques and tricks about changes of variables in both computation of limits and integrals

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17 views

Transforming random vector - checking correctness of solution

The task is to calculate two-dimentional probability density of $[X,Y]$ if we know that: $$ R \sim U(0,1), \quad \Phi \sim U(0, 2\pi), $$ $R$ and $\Phi$ are independent and $$ X := R \cos(\Phi), \...
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Using pdf vs using cdf to derive distribution of function of RV

I come across the following question: Let $X_1,...,X_n$ be a random sample from a distribution with density $f(x; β) = \frac{\beta}{(1+x)^{\beta+1}}$ when $x > 0$ and zero elsewhere. The parameter ...
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If a set $B$ can be covered in finitely many squares whose area is les any $\varepsilon$, then its the same for $\varphi(B)$ for some $C^1$ injection

On my way to proving change of variables theorem, there is a lemma that states: if $C \subset \mathbb R^2$ is an open, bounded set, and $\varphi : C \to \mathbb R^2$ is an injection of class $C^1$ on ...
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Change of variable for conditional probability

Suppose I have random variables $X$, $Y$ and $Z$, with $Z \sim N(0, \sigma^2)$ and $Y = kX + Z$, I am looking for a proof of the fact that $f_{Y\mid X}(y\mid X = x) = \frac{1}{\sqrt{2 \pi} \sigma} \...
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changing variables for numerical integration

I want to numerically integrate a set of data in Python, but I think my question might a math than a Python question. where I have discrete data points L, f(L) and I need to perform this integral: ...
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1answer
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Finding the area of an ellipse using change of coordinates

I would like to find the area of the ellipse $x^{2} +2xy +2y^{2} \leq 1$. I was told to use the substitution $s = x+y$ and $ t=y$. Using this, I found the Jacobian determinant to be $1$ and then ...
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1answer
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PDF of function of random variable in multidimensional case proof

I read the wiki page on PDF and I got stuck at the proof of relation between pdf $g$ of a function $\textbf{y}$ and pdf $f$ of its random variables $\textbf{x}$: $$g({\bf{y}}) = f({H^{ - 1}}({\bf{y}}...
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$ \text{why} -\int_{0}^{1}{({1 - t})^{n} - 1 \over t}d t = \int_{0}^{1}{t^{n} - 1 \over t - 1}d t ?$

Question link : Proving Binomial Identity without calculus i have one doubt in the given answer below ,my doubts mark in red colour My doubt is that $$ \text{why} -\int_{0}^{1}{({1 - t})^{n} - ...
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Change of variables on a double summation to yield a single sum

I'm going through a proof in Monson's Statistical Digital Signal Processing and Modelling on page 98. They used the substitution $k=n-m$ in order to change the double summation into a single summation....
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Help on calculating this integral and changing variables

Calculate the integral $I=\iint_D e^\frac{x^3+y^3}{xy}dA$ where D is the surface bounded by $x=y^2$,$x^2=y$ and $x^2=2y$. Attempt: Let $u=\frac{x^2}{y}$ and $v=\frac{y^2}{x}$. Then we have $1≤u≤2$ ...
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Simple Change of Variables in Lebesgue Integration

I'm not sure if these details matter, but anyway for this particular case, consider a compact abelian group $G$ with operation $\cdot$, a Haar measure $\mu$ on it and $f$ a non-trivial character on $G$...
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Change of integration variables across two maps in space and time

Question: Working on a numerical analysis problem involving continuum mechanics of fluids. How does the change of variables work when there are multiple maps involved? Context: In the following, ...
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Polar coordinates and Jacobian

Let $(V,W)$ a point in the circle of unity radius chosen in accordance with the following rules. First, let $R$ a random number uniform in $(0,1)$. Second, you choose a point $X$ on the circumference ...
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what is the distribution of $XY/Z$?

If $X$, $Y$, and $Z$ are independent random variables, each uniformly distributed over (0, 1), what is the distribution of $XY/Z$? I want to solve this exercises by the transformation method, The ...
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1answer
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How to solve $\int_0^2 \int_0^\sqrt{4-x^{2}} \int_0^\sqrt{4-x^2 -y^2} z \sqrt{4-x^2 -y^2} \, dz \, dy \, dx$ in spherical coordinate

$$\int_0^2 \int_0^\sqrt{4-x^{2}} \int_0^\sqrt{4-x^2 -y^2} z \sqrt{4-x^2 -y^2} \, dz \, dy \, dx$$ The task is to solve this integral using spherical coordinate. After I tried to change the variable, ...
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Limits of integration for new variables

I want to calculate an integral of some function on $V$ where $V$ is bounded by $x=0, y=0, z=0, z=1, 0\leq x^2+y^2 \leq 4, 1 \leq x^2 - y^2$ Using the change of variables $u=z, v = x^2+y^2, w = x^2 - ...
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Change of variable in integral, from real to complex variable

I have an integral depending on the following expression $$-(k_{20}-k_{10})^2 + (k_{2x}-k_{1x})^2 + (K_{2z}-K_1)^2 - K_2^2$$ where the integration variables are $k_{20}$ and $K_{2z}$. I need ...
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The volume of the image of a map with vanishing Jacobian is zero

Let $\Omega \subseteq \mathbb{R}^n$ be a nice domain with smooth boundary (say a ball), and let $f:\Omega \to \mathbb{R}^n$ be smooth. Set $\Omega_0=\{ x \in \Omega \, | \, \det df_x =0 \} $ Is ...
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Pdf upon transformation of multivariate random variable

What is the multivariate analogue to point 3. in this theorem? is || in here referring to absolute value? Thanks.
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Change the equation with new variables.

Recently at our school we was asked to try solve the following (our calculus course is rather basic so this kind of tasks is out of our scope, however the were considered as interesting and useful): ...
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Question About Changing Variable From Cartesian To Polar [closed]

Evaluate $$\int \int_R e^{-(x^2+y^2)} dxdy$$ Where R is the annulus bounded by $$y^2+x^2 = 1$$ and $$y^2+x^2=4$$ Changing into polar co-ordinates
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Calculating a surface integral using change of coordinates.

I need to use $u=x+y,\ v=x^2+y^2$ to find $I=\int_D x+y\ dxdy$ where $D=\{ (x,y)\ :\ x^2+y^2=1, y\geq 0\}$. I think the region for the $u-v$ plane is the rectangle $\{(u,v)\ :\ -1\leq u\leq \frac{1}{...
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Getting stuck at determining bounds

I think I know how to finish this problem, I just can't seem to figure out how to get the bounds for my substitution. Do you have any tips.
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1answer
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Finding the distribution of $\sqrt{X^2+Y^2}$

The joint pdf of $(X,Y)$ is given by: $f_{X,Y}(x,y)=4xye^{-(x^2+y^2)}$ if $0<x,y<\infty$ I have to find the distribution of $R=\sqrt{X^2+Y^2}$. I have approached this problem in the following ...
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Polar Coordinate Change of Variable when Domain is a Sphere

Evans computes by change of variables $z = x + sw$: $$\int_{\partial B(0,1)} |u(x+sw)-u(x)|\,dS(w) = \frac{1}{s^{n-1}}\int_{\partial B(x,s)} |u(z)-u(x)|\,dS(z)$$ The power $n-1$ confuses me. Can ...
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1answer
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Finding the distribution of $X-Y$ where $X$ and $Y$ are dependent random variables

The joint pdf of $(X,Y)$ is given by, $f_{X,Y}(x,y)=1$ if $0<x<2,0<y<1,2y\leq x$ Find the pdf of $U=X-Y$. My attempt: To solve the above, I consider the transformation $(X,Y)\...
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Let $(X,Y)$ be a 2D stochastic variable with density function $f(x,y) = 3/2 xy1_A(x,y)$. Find the distribution of $(X + Y)$ and $(X - Y)$.

Let $(X,Y)$, a 2D stochastic variable with density function $f(x,y) = 3/2 xy1_A(x,y)$, where $A$ is the set of all positive values bound by the line $y = 2-x$. Find the distribution of $(X + Y)$ and $...
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1answer
61 views

Sum $\frac{1}{2}\sum_{\substack{n = 1 \\n \text{ odd}}}^{\infty} \sum_{k=0}^n \frac{x^k}{k!} \frac{x^{n-k}}{(n-k)!}$

I find that every so often I end up with a double sum that could be changed into a different form and make some expression simpler. I have a situation I am in right now that should be able to benefit ...
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Laplace Equation in Polar Coordinates without $\frac{1}{R} \partial_R$

As I was reading a book on queue theory, the author needed to solve $p(V_b, V_a)$ where: $p(V_b, V_a) = \frac{1}{4}p(V_b+1, V_a) + \frac{1}{4}p(V_b, V_a+1) + \frac{1}{4}p(V_b-1, V_a) + \frac{1}{4}p(...
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Trying to to solve a limit by change of variables: legit or not.

I was trying to solve this limit: $$\lim_{x^2+y^2\to+\infty\\\; x\ge0,\;y\ge0}\frac{x^2y^3+\sin(x^2y)}{1+x^4+|y|^7}$$ 1.$\frac{x^2y^3+\sin(x^2y)}{1+x^4+|y|^7} \le \frac{x^2y^3+1}{1+x^4+|y|^7}$ $\qquad$...
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Finding the pdf of $U=X+Y$

$(X,Y)$ has the following joint pdf: $f_{X,Y}(x,y)=x+y$ if $0<x<1, 0<y<1$ If $U=X+Y$, find the marginal pdf of $U$. I have tried to do it using transformation. I have considered the ...
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1answer
26 views

Change of Variables Formula for Double Integral

In working on an example of the Change of Variables Formula, I seem to have hit a wall in understanding how to set up an integral. For the following, I have found the correct Jacobian, but am unsure ...
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1answer
102 views

Evaluating $\iint_{[0,1]^2} \frac{2-4xy}{(9-xy)(8+xy)}dxdy$

I am trying to compute the following double integral: $$I=\iint_S \frac{2-4xy}{(9-xy)(8+xy)}dxdy$$ with $S=[0,1]\times[0,1].$ What I have tried: I have written the integral as follows: $$I=I_1+I_2=-...
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1answer
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Continuous Random Variable Transformations vs Discrete

My Textbook, Introduction to Mathematical Statistics, has the following example of finding the pdf of a transformation of a continuous random variable: Let $X$ be a random variable with pdf $f_X(x)=...
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Calculate the Jacobian of a particular diffeomorphism of parallelizable manifold onto itself

Let $M$ be $d$-dimensional parallelizable manifold. Let $e_k(x)$ $k=1, \dots , d$ be the smooth vector fields forming orthonormal basis in tangent bundle of $M$, these vector fields exist because of ...
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References on Integration on non Compact manifolds

I am looking for references on integration on non-compact Riemannian manifolds, specially on the change of variables theorem. In particular I have non-compact manifold $M$ and I have an integral (in ...
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1answer
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Variable transformation of a Dirac delta function

I am struggling to understand the variable transformation of a Dirac delta function. More specifically, a transformation of the following type, $$\delta(a\chi(z)-b) \rightarrow \delta(z-c)$$ Here, $a, ...
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1answer
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Confusion about calculating integral with $d(-x)$

I'm wondering if $\int_{a}^{b}g(z)d(-z)=-\int_{a}^{b}g(z)dz$. On the one hand, intuitively this seems true as this looks like we are putting a negative sign on the $\Delta z$ in the Rieman sum, so it ...
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Correct Notation for change of variable

I am trying to clearing distinguish these cases below that have different meaning in the way the variables are used, yet the notation is similar. What are the preferred approaches to not have the ...
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1answer
55 views

Change of variable in two variables differential equation

I have a problem in understanding a passage from the nots of a professor of us. The starting problem is this PDE: $$ \dfrac{\partial^2 t}{\partial u^2} = \frac{1}{h}\frac{\partial}{\partial h}\left(...
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Calculate the volume between these surfaces…

In this exercise, I need to calculate the volume between these surfaces: Let $R>0$: The cylinder: $x^2+y^2-2Ry=0$ The paraboloid: $z=2R^2-x^2-y^2$ The plain: $z=0$ I'm stuck because the cylinder ...
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1answer
29 views

How to show that these sums are equal?

Assuming that the product is associative, I would like to show that $$ \sum_{s=0}^{k} (\sum_{i=0}^{s} (\alpha_i \cdot \beta_{s-i}) \cdot \gamma_{k-s}) = \sum_{s=0}^{k} ( \sum_{i=0}^{k-s} \alpha_s \...
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Probability density function of $X = \frac{P}{\cos(\theta)} - y\tan(\theta)$.

Assume that we have two independent random variables: $P$ is distributed on $(0,1)$ with $f_{P}(p) = 2p$. $\theta$ is uniformly distributed on $(0,2\pi)$. We then define $X,Y$ such that \begin{align*...
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1answer
60 views

Change of coordinates problem

Consider the region D defined by $1 \leq x^2-y^2 \leq 4$ and $ 0 \leq y \leq \frac{3x}{5}$. In the problem, set up an integral to compute $\int\int_{D} e^{x^2-y^2} dA$. Consider the change of ...
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A Double Integral Substitution

I am asked to compute the integral $$\int_\Omega xye^{x^2-y^2}dxdy$$ over the domain $\Omega = \{(x,y)\mid 1\leq x^2-y^2\leq 9, 0\leq x \leq 4, y\geq 0\}.$ After splitting the domain and a messy ...
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1answer
34 views

Distribution of $Z = \sin(X) \sin(Y)$ where $X$ and $Y$ are independent and uniform in $[-\pi,\pi]$?

Consider two random variables $X$ and $Y$ that are independent and uniformly distributed over a period, say $[-\pi,\pi]$. Which is the PDF (or the CDF if you prefer) of $Z = \sin(X) \sin(Y)$? This ...
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22 views

Changing a two variable limit to a one variable limit

Suppose we have the following limit, and assume it exists:$$\lim _{x\rightarrow a}\lim _{y\rightarrow b} f( x,y) =L$$ Then suppose we can find a function $g$ such that $$\lim _{x\rightarrow a} g( x) =...
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1answer
31 views

Probability Density Function from Random Variable Transformation

The problem 2.7 from Casella & Berger asks us to find the pdf of $Y = X^2$, given $$f_X(x)=\frac{2}{9}(x+1) \quad \text{ for }-1 \le x \le 2$$ I reasoned and concluded that, assuming $Y=g(X)=X^2$,...
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1answer
23 views

$X$ Cauchy distributed then $Y=\frac1X$ also Cauchy distributed

If $X$ is Cauchy distributed then $Y:=\frac1X$ is also Cauchy distributed. It holds: $$y=g(x)=\frac{1}{x} \Rightarrow g^{-1}(y)=\frac{1}{y} \Rightarrow (g^{-1}(y))'= -\frac{1}{y^2} $$ How does ...
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32 views

Integrate special function

I have the following integral $$ \int_a^b e^{\alpha_1 x + \alpha_2 x^2 + \alpha_3 e^{-\beta x}} dx, \\ \text{where} \, \alpha_1, \alpha_2, \alpha_3 \,\text{and} \, \beta \, \text{are constants} $$ I´...

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