Area Under a Linear Density Function
Given the function \(f(x) = kx\) on the interval \(0 \le x \le 2\) and \(f(x) = 0\) elsewhere, determine the value of \(k\) such that \(f(x)\) is a probability density function. Then, find the probability that a random variable \(X\) with this density lies in the interval \([1, 2]\).
Normalization of a Vector in Coordinate Geometry
Given the vector \(\vec{v} = (3, 4)\), find its corresponding unit vector.
Validating a Probability Mass Function
Given a discrete random variable X with possible values 1, 2, 3 and a probability mass function defined by \( P(X=x) = \begin{cases} a, & x=1 \\ 2a, & x=2 \\ 1-3a, & x=3 \end{cases} \) Determine the value of parameter a that makes this a valid pmf.
Calculating the Shortest Distance from a Point to a Line in 3D
A line L in three-dimensional space is given by its vector equation \(\mathbf{r} = (1, 4, 0) + t(2, -2, 2)\), where \(t\) is a real parameter. Given the point \(P(2, -1, 3)\), calculate the shortest distance from the point \(P\) to the line \(L\). Express your answer in simplest radical form.
Normalization of a 2D Vector
Given the vector \(\vec{v} = \langle 3, 4 \rangle\), find the unit vector in the direction of \(\vec{v}\).
Steady-State Analysis of a 2×2 Markov Chain
Consider a system with two states where the transition probabilities are given as follows: From state 1, the probability of remaining in state 1 is 0.7 and transitioning to state 2 is 0.3. From state 2, the probability of transitioning to state 1 is 0.4 and remaining in state 2 is 0.6. Work through the […]
Orthogonality in Real Symmetric Matrices
Given a real symmetric matrix \(A\), explain why eigenvectors corresponding to distinct eigenvalues are orthogonal, and briefly discuss how normalizing these eigenvectors leads to an orthonormal set.
Eigenvalue Analysis for a 2×2 Matrix
Consider the matrix \(A = \begin{pmatrix} 3 & 2 \\ 2 & 3 \end{pmatrix}\). (a) Determine the eigenvalues \(\lambda\) of \(A\) by solving the characteristic equation. (b) For the eigenvalue corresponding to the largest value found in part (a), find an eigenvector \(v\) associated with it. (c) Normalize the eigenvector obtained in part (b). (d) […]
Normalization of an Eigenvector
Given an eigenvector \(v = \begin{bmatrix} 3 \\ 4 \end{bmatrix}\) corresponding to some eigenvalue, scale \(v\) to unit length. Provide the normalized vector.