From 3189c28e2f972e3c66e7d10734826594800bb04e Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?=C5=81ukasz?= <lukasz.dudzinski1@gmail.com>
Date: Fri, 15 Apr 2016 10:02:22 +0200
Subject: [PATCH] Update zad_10.tex

---
 tydzien_06/zad_10.tex | 30 ++++++++++++++++++++++++++++++
 1 file changed, 30 insertions(+)

diff --git a/tydzien_06/zad_10.tex b/tydzien_06/zad_10.tex
index e69de29..791e285 100644
--- a/tydzien_06/zad_10.tex
+++ b/tydzien_06/zad_10.tex
@@ -0,0 +1,30 @@
+\subsection{Zadanie 10}
+$$
+Zmienna losowa $X$ ma gęstość 
+$$
+
+$$
+f(x)
+ = \left\{ \begin{array}{ll}
+\alpha \cdot \sqrt{x} & \textrm{gdy $x \in [0,1]$}\\
+0 & \textrm{gdy $x \notin [0,1]$}
+\end{array} \right.
+$$
+Oblicz $E(X)$.
+$$
+
+$$
+f(x)
+ = \left\{ \begin{array}{ll}
+\alpha \cdot \sqrt{x} & \textrm{gdy $x \in [0,1]$}\\
+0 & \textrm{gdy $x \notin [0,1]$}
+\end{array} \right.
+$$
+
+$$
+1 = \int_{0}^{\infty} \frac{\alpha}{x^3} dx = \alpha \int_{0}^{\infty} \frac{1}{x^3} dx = \alpha(-2) * \frac{1}{x^2} \right|_1^\infty = -2\alpha(\lim_{x\to\infty} \frac{1}{x^2} - 1) => 2\alpha = 1 <=> \alpha = \frac{1}{2}
+$$
+
+$$
+Ex = \int_{1}^{\infty} x * \frac{1}{2 * x^2} dx = \frac{1}{2} \int_{1}^{\infty} x^(-2) dx = \frac{1}{2} * (-1) * \frac{\alpha}{x} \right|_1^\infty = -\frac{1}{2} (\lim_{x\to\infty} \frac{1}{x} - 1) = \frac{1}{2}
+$$