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FUNDAMENTAL CONCEPTS
Understand the Relationship Between Probability and Statistics
Statistics is akin to reverse engineering probability to find the truth
Until a long time, I assumed that the expected value of a random variable is a fancy name for the sample mean. I was completely wrong.
Both the sample mean and the expected value are averages but they are not the same thing.
Let me explain it with a simple example given by Gilbert Strang, in one of his recent books (Linear Algebra and Learning from Data).
Imagine a class of fishermen. This is a class with 20% 17 years old, 50% 18 years old, and 30% 19 years old. Imagine picking a random sample of five fishermen aged 18, 17, 18, 19, and 17. The sample mean is 17.8.
However, the expected age of a randomly selected fisherman is 18.1.
Both 17.1 and 18.1, albeit different, are correct averages.
Understanding the difference between the two is in the name. The sample mean comes from a completed trial. It’s based on data that has already been collected.
The expected value comes from probabilities. It’s what we expect we will get if we undertake the trial.
