3.6.4 Normal Approximation of Discrete Random Variables

Question 1

What’s the big idea behind using the normal distribution for discrete variables?

Answer 1

If you’re summing a large number of independent discrete random variables, the Central Limit Theorem says the sum is approximately normal.

This allows for the use of normal distribution to approximate probabilities.

Question 2

Why do we need a continuity correction when using the normal distribution for a discrete variable?

Answer 2

Because the normal distribution is smooth and continuous while discrete distributions are chunky and jump between integers.

The continuity correction smooths out the chunkiness by adding/subtracting 0.5.

Question 3

What is the continuity correction for P(X = k)?

Answer 3

Approximate it as P(k - 0.5 < X < k + 0.5) using the normal distribution.

This correction accounts for the discrete nature of the variable.

Question 4

What are the steps to apply normal approximation to a discrete variable?

Answer 4

1. Find the mean and variance of the discrete variable. 2. Apply continuity correction. 3. Standardize to Z using Z = (X - μ) / σ. 4. Use the standard normal table.

Following these steps ensures accurate approximation of probabilities.