Stats Approximation of discrete distrubtions normal distribu

Stats: Approximation of discrete distrubtions, normal distributions, and central limit theorem.

Solution

Given that MGF = M(t) = e^{t+2t^2}

As MGF is exponential, this is a normal distribution

M\'(t) = (1+4t)e^{t+2t^2}

M\"(t) = (1+4t)²e^{t+2t^2}+4e^{t+2t^2}

Substitute 0 for t to get

M\'(0) = 1 and M\"(0) = 1+4 = 5

Or Mean = 1 and variance = 5

t is normal (1,rt 5)

P(0<x<3) = P(-0.45<Z<0.89)

= 0.1736+0.3106

= 0.4842