Expected Value Shift Transformation
Let \(X\) be a discrete random variable such that \(P(X = k) = \frac{1}{4}\) for \(k = 1, 2, 3, 4\). Calculate \(E(X+5)\).
Piecewise PMF Transformation and Expectation
Let X be a discrete random variable defined by the following probability mass function: – For \(x = 0\), \(p(x) = 0.2\) – For \(x = 1\), \(p(x) = 0.5\) – For \(x = 3\), \(p(x) = 0.3\) (a) Compute the expected value \(E(X)\) of \(X\). (b) If a new random variable \(Y\) is defined […]
Linear Transformations: Effects on E(X) and Var(X)
Let \(X\) be a random variable with \(E(X) = \mu\) and \(Var(X) = \sigma^2\). If a linear transformation \(Y = aX + b\) is applied, where \(a\) and \(b\) are constants, express \(E(Y)\) and \(Var(Y)\) in terms of \(\mu, \sigma^2, a,\) and \(b\).
Matrix Inversion and Determinant Evaluation
Let matrix \(M = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}\). (a) Find the inverse of matrix \(M\), denoted \(M^{-1}\). (b) Evaluate the determinant of \(M\) and interpret its meaning in terms of scaling and orientation of the geometric transformation represented by \(M\).