Let X have the following pdf
Let X have the following pdf
$$ f_{X}(x)=\left\{\begin{array}{rr} \frac{1}{2} & -1<x<1 \\ 0 & \text { otherwise } \end{array}\right. $$
1. Find the pdf of U=|X|.
2. Find the pdf of W=X3.
Homework Answers
3. Let X has the following pdf: {. -1 <1 fx(a) otherwise 1. Find the pdf...
3. Let X has the following pdf: {. -1 <1 fx(a) otherwise 1. Find the pdf of U X2. 2. Find the pdf of W X
I have the first method complete, but I can't figure out thesecond method. I have...
I have the first method complete, but I can't figure out the
second method Could
someone please show how to use the second method?2. Find the unit step response of:$$ \begin{aligned} \dot{\overrightarrow{\mathbf{x}}}(t) &=\left[\begin{array}{cc} 0 & 1 \\ -2 & -2 \end{array}\right] \overrightarrow{\mathbf{x}}(t)+\left[\begin{array}{l} 1 \\ 1 \end{array}\right] u(t) \\ y(t) &=\left[\begin{array}{cc} 2 & 3 \end{array}\right] \overrightarrow{\mathbf{x}}(t) \end{aligned} $$by two methods (1): transfer function and then (2) \(y(t)=\mathbf{C} e^{\mathbf{A} t} \overrightarrow{\mathbf{x}}(0)+\mathbf{C} \int_{0}^{t} e^{\mathbf{A}(t-\tau)} \mathbf{B} u(\tau) d \tau+\mathbf{D} u(t)\). Re-member that the Laplace...
Random variables X and Y have joint probability mass function (PMF):
Random variables \(X\) and \(Y\) have joint probability mass function (PMF):\(P_{X, Y}\left(x_{k}, y_{j}\right)=P\left(X=x_{k}, Y=y_{j}\right)= \begin{cases}\frac{1}{20}\left|x_{k}+y_{j}\right|, & x_{k}=-1,0,1 ; y_{j}=-3,0,3 \\ 0, & \text { otherwise }\end{cases}\)(a) Find \(F_{X, Y}(x, y)\), the joint cumulative distribution function (CDF) of \(X\) and \(Y\). A graphical representation is sufficient: probably the best way to do this is to draw the \(x-y\) plane and label different regions with the value of \(F_{X, Y}(x, y)\) in that region.(b) Let \(Z=X^{2}+Y^{2}\). Find the probability mass function (PMF)...
5. The amplitudes of two signals X and Y have joint pdf: Ísy(x,y)-kx(1-x)y, x, y є (0, 1). for (i...
5. The amplitudes of two signals \(X\) and \(Y\) have joint pdf:$$ f_{X, Y}(x, y)=k x(1-x) y, \text { for } x, y \in(0,1) \text { . } $$(i) Find the \(k\) and the joint CDF.(ii) Find the marginal pdfs and CDFs.(iii) Find \(P\left[Y<X^{1 / 2}\right]\).
Consider the linear system A.r = b where A = 1
Consider the linear system \(A x=b\) where \(A=\left[\begin{array}{rr}2 & -1 \\ -1 & 2\end{array}\right], b=\left[\begin{array}{l}1 \\ 1\end{array}\right], x=\left[\begin{array}{l}1 \\ 1\end{array}\right]\).We showed in class, using the eigenvlaues and eigenvectors of the iteration matrix \(M_{G S}\), that for \(x^{(0)}=\left[\begin{array}{ll}0 & 0\end{array}\right]^{T}\) the error at the \(k^{t h}\) step of the Gauss-Seidel iteration is given by$$ e^{(k)}=\left(\frac{1}{4}\right)^{k}\left[\begin{array}{l} 2 \\ 1 \end{array}\right] $$for \(k \geq 1\). Following the same procedure, derive an analogous expression for the error in Jacobi's method for the same system.
Consider the heat conduction problem
5. If \(f(x)=\left\{\begin{array}{cc}0 & -2<x<0 \\ x & 0<x<2\end{array} \quad\right.\)is periodio of period 4 , and whose Fourier series is given by \(\frac{a_{0}}{2}+\sum_{n=1}^{2}\left[a_{n} \cos \left(\frac{n \pi}{2} x\right)+b_{n} \sin \left(\frac{n \pi}{2} x\right)\right], \quad\) find \(a_{n}\)A. \(\frac{2}{n^{2} \pi^{2}}\)B. \(\frac{(-1)^{n}-1}{n^{2} \pi^{2}}\)C. \(\frac{4}{n^{2} \pi^{2}}\)D. \(\frac{2}{n \pi}\)\(\mathbf{E}_{1} \frac{2\left((-1)^{n}-1\right)}{n^{2} \pi^{2}}\)F. \(\frac{4}{n \pi}\)6. Let \(f(x)-2 x-l\) on \([0,2]\). The Fourier sine series for \(f(x)\) is \(\sum_{w}^{n} b_{n} \sin \left(\frac{n \pi}{2} x\right)\), What is \(b, ?\)A. \(\frac{4}{3 \pi}\)B. \(\frac{2}{\pi}\)C. \(\frac{4}{\pi}\)D. \(\frac{-4}{3 \pi}\)E. \(\frac{-2}{\pi}\)F. \(\frac{-4}{\pi}\)7. Let \(f(x)\) be periodic...
Suppose that
Suppose that \(\left(\xi_{j}\right)^{\infty}=1\) is a sequence of independent identically distributed \((i . i . d .)\) continuous random variables.- Suppose that each \(\xi_{i}\) has a probability density function \(p_{i}(x)=\left\{\begin{array}{c}\frac{\beta}{x^{x}}, x \geq 1 \\ 0, x<1\end{array}\right.\), where \(\alpha, \beta \in\)R.- Let \(S_{n}=\sum_{i=1}^{n} \xi_{i}\).- Let \(S_{n}=\frac{s_{n}-\operatorname{mE}\left(\xi_{0}\right)}{\sqrt{n} \operatorname{Var}(\mathcal{B})}\).a. Find a condition on \(\alpha\) and a condition on \(\beta\) (as a function of \(\alpha\) ) which together make \(f_{1}(x)\) a probability density function.b. Find conditions on \(\alpha\) which guarantee that \(\lim _{n \rightarrow \infty}...
Assess a subjective triangular probability distribution for the random variable, X, where X is defined as...
Assess a subjective triangular probability distribution for the
random variable, X, where X is defined as the amount of snowfall
you think we will get in our next snowstorm in inches. Suppose that
you expect your commute to increase in minutes according to the
function . What is the expected increase in your
commute for the next snowstorm?
Estimate E[f(X)] by creating simple Monte Carlo
simulation in Excel.
Here are some equations for the triangular distribution that may
help you:...
MATHEMATICS - III
$$
\begin{array}{l}
\sum_{r=0}^{n} T_{2 r}(x)=\frac{1}{2}\left(1+\frac{1}{\left(1-x^{2}\right)^{1 / 2}} U_{2 n+1}(x)\right) \\
\text { where, } U_{n}(x)=\sin \left(n \cos ^{-1} x\right) \text { and } T_{n}(x)=\cos \left(n \cos ^{-1} x\right)
\end{array}
$$
Defining the cross product The cross product of two nonzero vectors
Defining the cross product The cross product of two nonzero vectors \(\vec{u}\) and \(\vec{v}\) is another vector \(\vec{u} \times \vec{v}\) with magnitude$$ |\vec{u} \times \vec{v}|=|\vec{u}||\vec{v}| \sin (\theta), $$where \(0 \leq \theta \leq \pi\) is the angle between the two vectors. The direction of \(\vec{u} \times \vec{v}\) is given by the right hand rule: when you put the vectors tail to tail and let the fingers of your right hand curl from \(\vec{u}\) to \(\vec{v}\) the direction of \(\vec{u} \times \vec{v}\)...
3. Let X has the following pdf: {. -1 <1 fx(a) otherwise 1. Find the pdf of U X2. 2. Find the pdf of W X
I have the first method complete, but I can't figure out the
second method Could
someone please show how to use the second method?2. Find the unit step response of:$$ \begin{aligned} \dot{\overrightarrow{\mathbf{x}}}(t) &=\left[\begin{array}{cc} 0 & 1 \\ -2 & -2 \end{array}\right] \overrightarrow{\mathbf{x}}(t)+\left[\begin{array}{l} 1 \\ 1 \end{array}\right] u(t) \\ y(t) &=\left[\begin{array}{cc} 2 & 3 \end{array}\right] \overrightarrow{\mathbf{x}}(t) \end{aligned} $$by two methods (1): transfer function and then (2) \(y(t)=\mathbf{C} e^{\mathbf{A} t} \overrightarrow{\mathbf{x}}(0)+\mathbf{C} \int_{0}^{t} e^{\mathbf{A}(t-\tau)} \mathbf{B} u(\tau) d \tau+\mathbf{D} u(t)\). Re-member that the Laplace...
5. The amplitudes of two signals \(X\) and \(Y\) have joint pdf:$$ f_{X, Y}(x, y)=k x(1-x) y, \text { for } x, y \in(0,1) \text { . } $$(i) Find the \(k\) and the joint CDF.(ii) Find the marginal pdfs and CDFs.(iii) Find \(P\left[Y<X^{1 / 2}\right]\).
Assess a subjective triangular probability distribution for the
random variable, X, where X is defined as the amount of snowfall
you think we will get in our next snowstorm in inches. Suppose that
you expect your commute to increase in minutes according to the
function . What is the expected increase in your
commute for the next snowstorm?
Estimate E[f(X)] by creating simple Monte Carlo
simulation in Excel.
Here are some equations for the triangular distribution that may
help you:...
Defining the cross product The cross product of two nonzero vectors \(\vec{u}\) and \(\vec{v}\) is another vector \(\vec{u} \times \vec{v}\) with magnitude$$ |\vec{u} \times \vec{v}|=|\vec{u}||\vec{v}| \sin (\theta), $$where \(0 \leq \theta \leq \pi\) is the angle between the two vectors. The direction of \(\vec{u} \times \vec{v}\) is given by the right hand rule: when you put the vectors tail to tail and let the fingers of your right hand curl from \(\vec{u}\) to \(\vec{v}\) the direction of \(\vec{u} \times \vec{v}\)...
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