Probability Questions (Chebychev's Bound) Suppose we want to estimate the pH of a mysterious liquid. Let the pH be 4. We take $t$ readings and let $X_i$ be the value returned by the ith reading. You should assume that the readings are independent and that $E(X_i) = 4$ and $var (X_i) = 3$. Let $Y = (X_1 + X_2 + \cdots + X_t)/t$ be the average of these readings.
How large does $t$ need to be such that $P(3.9 < Y < 4.1) \geq 0.99$. Apparently we can use Chebychev's bound
Suppose the equipment you use for reading the pH only outputs the values 2,4, or 6. Given that the expected reading is 4 and the variance is 3, what is the probability that the reading takes each of the three possible values?
I know the formula for Chebychev's inequality, but I have no idea how to do these problems. any tips?
 A: The version of Chebyshev's inequality you want to use is probably the following one:
$$ P(|Y-\mu|<\epsilon)\geq 1-\frac{\sigma^2}{\epsilon^2},$$
where $\mu=E(Y)$ and $\sigma^2=Var(Y)$. Now, rearranging the terms in your probability, you get
$$P(\mu-\epsilon<Y<\mu +\epsilon)\geq 1-\frac{\sigma^2}{\epsilon^2}.$$
Now, all there is left to do is to compute $\mu$ and $\sigma^2$ (Hint: $\sigma^2$ will depend on $t$, thus the link between your problem and Chebyshev's inequality), and choose an appropriate $\epsilon$.
As for your second problem, you know your random variable is discrete, taking only three values; call these probabilities $p_2$, $p_4$ and $p_6$. From the definition of the expected value, you have 
$$E(X_i)=2p_2+4p_4+6p_6.$$
Since you're given the value of this expectation, you are left with an equation involving three unknowns. Similarly, the variance will give you a second equation involving the same three unknowns. Can you find a third equation? (Hint: what is the sum of the $p_i$s?)
Edit: Let's be even more explicit about the second problem. By one of the axioms of probability, we have
$$p_2+p_4+p_6=1.$$
As noted above, we also have
$$2p_2+4p_4+6p_6=4.$$
Finally, since $E(X_i^2)=Var(X_i)+E(X_i)^2=19$, we have a third equation:
$$4p_2+16p_4+36p_6=19.$$
So what you have is a system of three equations in three unknowns. By solving this system, you will find the values of $p_2,p_4,p_6$.
