$X$ is the midterm grade (out of $100$ points) of a randomly chosen student. If the
average grade is $40$ points, find an upper bound on $Pr[X < 10]$ using Markov's
inequality. (Hint: You may need to define a new non-negative random variable.)
I encountered this question and applied Markov's inequality but it gave me
$P(X < 10)>= 1-\left(\frac{40}{10}\right)$ and that gives us a negative bound. How can I resolve this problem ?
Best Answer
The problem is that $P( X \leq 10) $ is not of the form desired by the Markov inequality, and using $P(X \leq 10) = 1- P(X \geq 10)$ followed by Markov does not help since the RHS becomes negative, so the inequality is obvious.
To get it to the Markov form, you must get a non-negative random variable $X'$ and a quantity $T > 0$ so that $P(X' \geq T) = P(X \leq 10)$. Then you can apply Markov inequality to $X'$ to conclude.
In this case, for example take $X' = 100-X$, then $E[X'] = 100-E[X]$ and $P(X' \geq 90) = P(X \leq 10)$. Use Markov inequality, conclude.