Question

From past experience, a professor knows that the test score of a student taking her final examination is a random variable with mean 70 and variance 8.  What is the upper bound of the probability that a student will score between 65 and 75

Answer 1

Chebyshev’s inequality states that for any probability distribution of an rv $X$ and any number $k$ that is at least $1,P(|X-\mu|\geq k\sigma)\leq 1/k^2.$

$$P(65\leq X\leq75)=P(|X-70|\leq 5)$$

$$=1-P(|X-70|>5)\geq 1-\dfrac{\sigma^2}{5}$$

$$=1-\dfrac{8}{5^2}=0.68$$