Question 1

Why is the variance $\sigma^2$ used in the notation $N(\mu, \sigma^2)$ instead of the standard deviation $\sigma$?

Accepted Answer

This is a mathematical convention. Variance has useful additive properties that standard deviation does not. For example, the sum of two independent normal variables has a variance that is the sum of their variances. Using variance in the notation makes these theoretical properties clearer.

Question 2

What do I do if my calculated z-value is not in the table?

Accepted Answer

The standard normal table in the Cambridge formula book (MF19/MF20) is quite comprehensive. If a value is between two listed values, you can perform linear interpolation to get a more accurate probability. For values larger than those in the main table (e.g., z > 3.5), the probability $P(Z < z)$ is very close to 1.

Question 3

How do I know when to use a continuity correction?

Accepted Answer

You must use a continuity correction whenever you are using a continuous distribution (like the normal distribution) to approximate a discrete distribution (like the binomial or Poisson distributions). If the original problem involves a variable that can only take integer values, a continuity correction is required for the approximation.

Question 4

Can I use the normal approximation for a Poisson distribution?

Accepted Answer

Yes. A Poisson distribution $Po(\lambda)$ can be approximated by a normal distribution $N(\lambda, \lambda)$ provided that $\lambda$ is large (typically $\lambda > 15$). Just like with the binomial approximation, you must use a continuity correction.

Question 5

What's the difference between $P(X < 5)$ and $P(X \le 5)$ for a normal distribution?

Accepted Answer

For any continuous distribution, including the normal distribution, the probability of the variable being exactly equal to a single value is zero. i.e., $P(X=5)=0$. Therefore, for a continuous variable $X$, $P(X < 5)$ is exactly the same as $P(X \le 5)$. This is NOT true for discrete distributions, which is why continuity corrections are so important when approximating.

The normal distribution

What you need to know

Decoding the Bell Curve