ThmDex – An index of mathematical definitions, results, and conjectures.
Result R3559 on D205: Probability distribution function
Probability of random real number taking value in right-closed interval
Formulation 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $X$
(ii) $F_X : \mathbb{R} \to [0, 1]$ is a D205: Probability distribution function for $X$
Let $a, b \in \mathbb{R}$ each be a D993: Real number such that
(i) \begin{equation} a \leq b \end{equation}
(ii) $(a, b] \subseteq \mathbb{R}$ is a D1278: Right-closed real interval
Then \begin{equation} \mathbb{P}(X \in (a, b]) = F_X(b) - F_X(a) \end{equation}
Formulation 1
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $X$
Let $a, b \in \mathbb{R}$ each be a D993: Real number such that
(i) \begin{equation} a \leq b \end{equation}
(ii) $(a, b] \subseteq \mathbb{R}$ is a D1278: Right-closed real interval
Then \begin{equation} \mathbb{P}(a < X \leq b) = \mathbb{P}(X \leq b) - \mathbb{P}(X \leq a) \end{equation}
Proofs
Proof 0
Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that
(i) $X : \Omega \to \mathbb{R}$ is a D3161: Random real number on $X$
Let $a, b \in \mathbb{R}$ each be a D993: Real number such that
(i) \begin{equation} a \leq b \end{equation}
(ii) $(a, b] \subseteq \mathbb{R}$ is a D1278: Right-closed real interval
We have \begin{equation} \begin{split} \{ X \leq b \} = \{ X \in (-\infty, b] \} & = \{ X \in (-\infty, a] \text{ or } x \in (a, b] \} \\ & = \{ X \leq a \text{ or } a < X \leq b \} \\ & = \{ X \leq a \} \cup \{ a < X \leq b \} \\ \end{split} \end{equation} Subtracting $\{ X \leq a \}$ from both sides, mapping both sides in $\mathbb{P}$, and applying R2060: Probability of set difference then gives \begin{equation} \mathbb{P}(a < X \leq b) = \mathbb{P}( \{ X \leq b \} \setminus \{ X \leq a \} ) = \mathbb{P}(X \leq b) - \mathbb{P}(X \leq a) \end{equation} $\square$