[Solved] Suppose X follows a distribution with the density function: f_X(x \mid r, λ)=(lambda^r)/(Gamma(r)) x^r-1 e^-λ x, x ≥q 0, λ>0,
Question: Suppose \(X\) follows a distribution with the density function:
\[f_{X}(x \mid r, \lambda)=\frac{\lambda^{r}}{\Gamma(r)} x^{r-1} e^{-\lambda x}, x \geq 0, \lambda>0, r>0\]- Show that its moment generating function is \(M_{X}(t)=\left(\frac{1}{1-t / \lambda}\right)^{r} .\) (6 points)
- Find \(\mathrm{E}(X)\) and \(\operatorname{Var}(X)\). (6 points)
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Let \(X\) and \(Y\) be independent random variables that follow density functions \(f_{X}\left(x \mid \frac{r}{2}, \frac{1}{2}\right)\) and \(f_{Y}\left(y \mid \frac{s}{2}, \frac{1}{2}\right)\), respectively; \(r\) and \(s\) are positive integers. Please note that the functions are both defined in the form of (*). Determine whether that
\(Z_{1}=X+Y\) and \(Z_{2}=X\) are independent. (10 points) - Identify the distribution of \(Z_{1}\) and find \(\mathrm{E}\left(Z_{1}\right)\) and \(\operatorname{Var}\left(Z_{1}\right)\) (4 points)
Deliverable: Word Document 
(Steps Shown) The amount (unit: ounce) of fill dispensed by a bottle
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[Solution Library] Compute the following limits or argue that they do
(Step-by-Step) Consider the integral ∫_0^1 ∫_1^e^x d y d
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[See Steps] Compute the integral ∫_0^2 ∫_0^√4-y^2
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