8.2.10
A student’s score on a particular calculus final is a random variable with values of [0, 100], mean 70, and variance 25.
To summarise: \(\mu = 70\) and \(\sigma^2 = 25\)
- Find a lower bound for the probability that the student’s score will fall between 65 and 75.
To find this, we use Chebyshev’s Inequality: \(P(|X-\mu| \ge \in ) \le \frac{\sigma^2}{\in^2}\), where \(k= \mbox{# students} =1\) and \(\in = k\sigma = 1*\sqrt{Var} = \sqrt{25} = 5\) \[P(|X-70| \ge 5) = \frac{25}{5^2} = 1 \] \[\therefore \mbox{the Lower Bounds} = 1 - P(|X-70| \ge 5) = 1-1 = 0\]
- If 100 students take the final, find a lower bound for the probability that the class average will fall between 65 and 75.
Again, we use Chebyshev’s Inequality: \(P(|X-\mu| \ge \in ) \le \frac{\sigma^2}{\in^2}\), where \(k= \mbox{# students} =100\) and \(\in = k\sigma = 100*\sqrt{Var} = \sqrt{25} = 500\) \[P(|X-70| \ge 5) = \frac{25}{500} = 0.01 \]
\[\therefore \mbox{the Lower Bounds} = 1 - P(|X-70| \ge 5) = 1-0.01 = 0.99\]