Use the Markov Inequality and with a carefully chosen value

Use the Markov Inequality and with a carefully chosen value of a to compute a bound on P [X= 6] .Choose a to make the bound as tight as possible with this inequality and compare your result to the exact value.

Let X be the number of dots on the top side of a randomly rolled fair die. Compute the mean and variance of X. Use the Markov Inequality with a carefully chosen value of a to compute a bound on P[X = 6]. Chose a to make the bound as tight as possible with this inequality and compare your result to the exact value. Use the Chebychev Inequality with a carefully chosen value of a to compute a bound on P[X = 6 or X = 1]. Chose a to make the bound as tight as possible with this inequality and compare your result to the exact value.

Solution

X is the no of dots on a fair die when it is thrown

As the die is fair and each throw is independent,

E(x) = (1+2+3+..+6) 1/6 = 21/6 = 3.5

E(X^2) = (1^2+2^2+...+6^2) 1/6 = 91/6

Var(X) = 91/6- 49/4

= (181-147)/12 = 17/6