Question

X is a random variable with Moment Generating Function M(t) = exp(3t + t²).

Calculate P[ X > 3 ]

Answer 1

The moment generating function given is:

M(t) = e^(3t + (t^2))

We know that the MGF of a normal distribution is given by:

MGF of normal distribution = e^(ut + 0.5**(t^2))

Comparing we get:

Mean, u = 3

Standard deviation, = 1.414

So the given mgf is of a normal distribution with mean equal to 3 and standard deviation equal to 1.414

So,

At X = 3, we have:

z = (X-u)/ = (3-3)/1.414 = 0

The corresponding p-value for this z-value is:

P(X > 3) = P(z > 0) = 0.5