Chapter 1: Problem 6
For each of the following pdfs of \(X\), find \(P(|X|<1)\) and
\(P\left(X^{2}<9\right)\).
(a) \(f(x)=x^{2} / 18,-3
Chapter 1: Problem 6
For each of the following pdfs of \(X\), find \(P(|X|<1)\) and
\(P\left(X^{2}<9\right)\).
(a) \(f(x)=x^{2} / 18,-3
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Get started for freeConsider \(k\) continuous-type distributions with the following characteristics: pdf \(f_{i}(x)\), mean \(\mu_{i}\), and variance \(\sigma_{i}^{2}, i=1,2, \ldots, k .\) If \(c_{i} \geq 0, i=1,2, \ldots, k\), and \(c_{1}+c_{2}+\cdots+c_{k}=1\), show that the mean and the variance of the distribution having pdf \(c_{1} f_{1}(x)+\cdots+c_{k} f_{k}(x)\) are \(\mu=\sum_{i=1}^{k} c_{i} \mu_{i}\) and \(\sigma^{2}=\sum_{i=1}^{k} c_{i}\left[\sigma_{i}^{2}+\left(\mu_{i}-\mu\right)^{2}\right]\) respectively.
Find the moments of the distribution that has mgf \(M(t)=(1-t)^{-3}, t<1\). Hint: Find the MacLaurin's series for \(M(t)\).
Find the pdf \(f(x)\), the 25 th percentile, and the 60 th percentile for each
of the following cdfs: Sketch the graphs of \(f(x)\) and \(F(x)\).
(a) \(F(x)=\left(1+e^{-x}\right)^{-1},-\infty
Let \(X\) have the uniform pdf \(f_{X}(x)=\frac{1}{\pi}\), for
\(-\frac{\pi}{2}
Our proof of Theorem \(1.8 .1\) was for the discrete case. The proof for the continuous case requires some advanced results in in analysis. If in addition, though, the function \(g(x)\) is one-to-one, show that the result is true for the continuous case. Hint: First assume that \(y=g(x)\) is strictly increasing. Then use the change of variable technique with Jacobian \(d x / d y\) on the integral \(\int_{x \in \mathcal{S}_{x}} g(x) f_{X}(x) d x\)
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