Question

Let X be a binomial random variable with parameters (n, p). Find the approximate value of P(|X| > a) when n is sufficiently large using the central limit theorem, where a is a positive number. g

Answer 1

Approximate value of P(|X| > a) is attached below

Ф ( 0 ) is the CDF of standard Normal variate

Explanation:

x ~ Bin ( n, p )

E ( x ) = np , Var ( x ) = np ( 1 - p )

∴ Z = Iin ( x - E(x) ) / √Var x ) ~ N ( 0, 1 ) ( this is the standard normal Variate)

n-1∞

Find the approximate value of P(|X| > a)

Ф ( 0 ) is the CDF of standard Normal variate

attached below is the remaining part of the solution