Probability of exceeding a value with a given expectation and variance?
A factory produces $n$ items a week and is a random variable with expectation $50$ and variance $25$ what can be said about the probability that this weeks production will exceed $75$? The question then asks; What can be said about the probability that this week’s production will be between $40$ and $60$?
I understand the formulas for $E(x) = \frac { (n_1 + n_2 + n_3 + ... + n_m)} {m}$
and $ Var(x) = E(x^2) - E (x)^2 $
I just cant figure out how to use these two values to use them to answer the questions? Any help or explanations will be much appreciated
Solution 1:
- you can use Markov's inequality
$$\mathbb{P}[X\ge 75]\le \frac{50}{75}$$
- Invoking Chebishev's inequality you get
$$\mathbb{P}\{|X-50|<10\}|\ge1- \frac{25}{10^2}=1-0.25$$