Answer to Question #159406 in Statistics and Probability for TasRak

Question

The hospitalization period, in days, for patients following treatment for a certain

type of virus X, where X has the density function f(x) = 4/{π(x² +1)}

, 0<x <58. Find

the expected value of X. that a person is hospitalized following treatment for this

disorder.

Answer 1

$$\displaystyle\int_{-\infin}^{\infin}f(x)dx=\displaystyle\int_{0}^{58}\dfrac{4}{\pi(x^2+1)}dx$$

$$=\dfrac{4}{\pi}[\arctan(x)]\begin{matrix} 58 \\ 0 \end{matrix}=\dfrac{4\arctan(58)}{\pi}\not=1$$

Therefore

$$f(x) = \begin{cases} \dfrac{1}{\arctan(58)\cdot(x^2+1)} &\text{if } 0<x<58 \\ 0 &\text{otherwise } \end{cases}$$

$$E(X)=\displaystyle\int_{-\infin}^{\infin}xf(x)dx$$

$$=\displaystyle\int_{0}^{58}x(\dfrac{1}{\arctan(58)(x^2+1)})dx$$

$$=\dfrac{1}{2\arctan(58)}[\ln(x^2+1)]\begin{matrix} 58 \\ 0 \end{matrix}=$$

$$=\dfrac{\ln(3365)}{2\arctan(58)}\approx2.613739$$