- #1

mathmari

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MHB

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The distribution of a random variable $X:\Omega\rightarrow \mathbb{R}$ is given by the probability function $p_X$ as follows:

(i) Calculate the distribution function $F_X$ of $X$.

(ii) Let $Y(\omega)=|X(\omega)|$ for all $\omega\in \Omega$. Determine $p_Y$. I have done the following :

(i) \begin{align*}&F_X(-4)=p_X(-4)=\frac{1}{4} \\ &F_X(0)=p_X(-4)+p_X(0)=\frac{1}{4}+\frac{1}{6}=\frac{5}{12} \\ &F_X(4)=p_X(-4)+p_X(0)+p_X(4)=\frac{1}{4}+\frac{1}{6}+\frac{1}{4}=\frac{5}{12}+\frac{1}{4}=\frac{2}{3} \\ &F_X(12)=p_X(-4)+p_X(0)+p_X(4)+p_X(12)=\frac{1}{4}+\frac{1}{6}+\frac{1}{4}+\frac{1}{3}=\frac{2}{3}+\frac{1}{3}=1\end{align*}

(ii) We have that $Y(\Omega)\in \{0, 4, 12\}$, right? But how can we calculate the probability? :unsure: