Exam 2.1 (consider the probability density function f(x) for the continuous random variable X for 1-6)

Define the cumulative distribution function F(x)

F(x) = P(X<=x)

How can you calculate the cumulative distribution function F(x)

F(x)= P(X<=x) = integral from -inf to x of f(x)dx

What are the mean and variance of a standard normal random variable Z?

Mean=E(Z) = mue = 0 VAR(Z) = 1

What is the probability P(X=0)

P(X=0)=0

How would you calculate the P(x1<=X<=x2)

P(x1<=x<=x2) = integral from x1 to x2 of f(x)dx

What is continuity correction, and what is it used for?

The continuity correction is a correction factor used to improve the approximation of binomial random distributions by normal distributions

What very important property is associated with the random variable X described in 1.3 above?

The memoryless

Explain what is meant by standardizing a normal random variable

To standardize a normal random variable X we let Z = (X-mue)/ sigma. Where Z is a standard normal random variable

Given F(x) how can you find f(x)

f(x) = d(F(x))/dx

Give the conditions that f(x) must satisfy

i) f(x)>=0 for all -inf<x<inf ii) f(x) is piece-wise continuous iii) integral from -inf to +inf of f(x)dx = 1

What do we call a random variable X that represents the distance between successive events from a Poisson process, with mean number of events lambda > 0, per unit

An exponential random variable