06. CDF of the normal distribution & the standard normal distribution

Understanding Normal Distribution and CDF

Key Concepts:

• Normal Distribution: A probability distribution that is symmetric around the mean, defined by mean (μ) and standard deviation (σ).
• Cumulative Distribution Function (CDF): Gives the probability that a random variable is less than or equal to a specific value. Expressed as \Phi (x), where x is the variable.
• Probability Density Function (PDF): Provides the likelihood of a random variable in a particular range.

Important Definitions:

• Standard Normal Distribution: A normal distribution with a mean of 0 and a standard deviation of 1.
• Transformation: For any real numbers a and b, if X is standard normal, then aX+b results in a normal distribution with mean b and standard deviation |a|.

Practical Example:

• For measuring IQ differences, use norm.cdf to find probabilities.
• Formula: norm.cdf(120,100,15)=norm.cdf(20/15) gives a CDF value determining probability.

Tools:

• Use scipy.stats in Python for calculations:
  • Import with from scipy.stats import norm.
  • Calculate using norm.pdf and norm.cdf functions.