upper bound | Problems in Mathematics

02/02/2020

Upper Bound of the Variance When a Random Variable is Bounded

Problem 753

Let $c$ be a fixed positive number. Let $X$ be a random variable that takes values only between $0$ and $c$. This implies the probability $P(0 \leq X \leq c) = 1$. Then prove the next inequality about the variance $V(X)$.
\[V(X) \leq \frac{c^2}{4}.\]

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Probability

01/17/2020

Lower and Upper Bounds of the Probability of the Intersection of Two Events

Problem 741

Let $A, B$ be events with probabilities $P(A)=2/5$, $P(B)=5/6$, respectively. Find the best lower and upper bound of the probability $P(A \cap B)$ of the intersection $A \cap B$. Namely, find real numbers $a, b$ such that
\[a \leq P(A \cap B) \leq b\] and $P(A \cap B)$ could take any values between $a$ and $b$.

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