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index 0d3f41f..7e79ed1 100644 --- a/ukazkove-pisemky.tex +++ b/ukazkove-pisemky.tex @@ -102,7 +102,7 @@ \begin{example} Náhodná veličina $X$ má spojité rozdělení s hustotou $$ f(x) = \begin{cases}2e^{-2x}, x \geq 0;\\0, \text{jinak}.\end{cases} $$ - Definujme $U := \floor{X}$ a $V := X - \floor{X}$ (tedy spodní celá část a frakcionální část $X$). + Definujme $U := \floor{X}$ a $V := X - \floor{X}$ (jinými slovy spodní celá část a frakcionální část $X$). \begin{enumerate}[(a)] \item Určete rozdělení náhodné veličiny $U$.\\ \textit{Jedná se o diskrétní náhodnou veličinu s pravděpodobnostní funkci @@ -125,10 +125,10 @@ $$f(x, y) = \begin{cases}2x, 0 < x < 1, -x^2 < y < x^2;\\0,\text{jinak}.\end{cases}$$ \begin{enumerate}[(a)] \item Určete $P(0 < 2Y < X^2)$.\\ - \textit{Počítáme obsah útvaru mezi osou $x$ a křivkou $y = \frac{1}{2}x^2$ s hustotou $f(x, y)$. Tedy máme integrál + \textit{Počítáme obsah útvaru mezi osou $x$ a křivkou $y = \frac{1}{2}x^2$ s hustotou $f(x, y)$. Máme integrál $$ \int_0^1 \int_0^{\frac{1}{2}x^2} 2x dy dx = \int_0^1 x^3 dx = \frac{1}{4}. $$} \item Určete marginální rozdělení a střední hodnotu veličiny $X$.\\ - \textit{Hustotu $f_X$ získáme zintegrování hustoty $f(x, y)$ pro všechny možné hodnoty $y$, tedy + \textit{Hustotu $f_X$ získáme zintegrováním hustoty $f(x, y)$ přes všechny možné hodnoty $y$, tedy $$ f_X(x) = \int_{-x^2}^{x^2} 2x dy = 4x^3. $$ Dále platí $$ \E X = \int_0^1 x f_X(x) dx = \int_0^1 4x^4 dx = \frac{4}{5}. $$}