Question

chebyshev’s inequality

Problem 2 Chebyshev's Inequality Suppose that the random variable ? has a Poisson distribution with the parameter ? > 0, ~ ?(A). Using the Chebyshev's inequality prove that Problem 3 - Application of the Chebyshev's Inequality Suppose that a player plays a game where he gains a dollar with the probability or loses a dollar with the probability . That is, his gain from one game can be modeled as a random variable fi, such that If the player plays independently n games, then his total gain will be S ?-14 Suppose that we want to determine his total gain by a probability greater than 0.8. How many games should the player play? After determining the number of games, use it to calculate approximately (give an interval) his total gain.

Answer 1

given X1,X2,...,Xn (zeta replaced by x for convenience) follow Bernoulli distribution with p = 7/8 with

$S_{n} = \sum_{i=1}^{n} X_{_{n}}$

therefore Sn follows binomial distribution(n,p=7/8).

E(Sn/n) = np/n = p =7/8

Var(Sn) = pq/n =(7/8*1/8)/n

$p[ \left | \frac{S_{n}}{n} -p \right | < k ] \geqslant 1-\frac{Var(S_{n}/n)}{k^{2}}$

choosing k =0.1,

we have

$1-\frac{Var(S_{n}/n)}{k^{2}} =0.8$

1-(pq/n)/(0.1*0.1) =0.8

n =7/0.128

n = 54.6875

hence n = 55

so he has to play at least n = 55 games to get gain = np = 55*(7/8) =48.125.