English explorer, Inventor, Anthropologist
(1822–1911)
He studied medicine and mathematics at Cambridge University.
He invented the phrase “nature versus nurture”
In his will, he donated funds for a professorship in genetics to University College London.
We take each ball independently
In every level, the ball bounces either left or right
We represent these options as -1 and 1
At the last level the position is the sum of all bounces
M
, larger varianceNow we have a coin 𝑋 with two possible outcomes: +1 and -1
To make life easy, we assume 𝑝=0.5
What are the expected value and variance of X ?
We throw the coin 𝑛 times, and we calculate 𝑌, the sum of all 𝑋 \[Y=\sum_{i=1}^𝑛 X_i\]
What are the expected value and variance of 𝑌 ?
Now consider \(Z_n=Y/\sqrt{𝑛}\)
It is easy to see that \(𝔼Z_n = 0\) and \(𝕍Z_n = 1\) independent of 𝑛
The possible values of \(Z_n\) are not integers. Not even rationals
What happens with \(Z_n\) when 𝑛 is really big?
This “bell-shaped” curve is found in many experiments, especially when they involve the sum of many small contributions
It is called Gaussian distribution, or also Normal distribution
Here outcomes are real numbers
Any real number is possible
Probability of any \(x\) is zero (!)
We look for probabilities of intervals
“The sum of several independent random variables converges to a Normal distribution”
The sum should have many terms, they should be independent, and they should have a well defined variance
(In Biology sometimes the variables are not independent, so be careful)
When \(n→∞,\) the distribution of \(Z_n=∑ X/\sqrt{𝑛}\) will converge to a Normal distribution \[\lim_{n→∞} Z_n ∼ Normal(0,1)\]
If \(X_i\) is a set of i.i.d. random variables, with \[𝔼X_i=μ\quad\text{and}\quad 𝕍X_i=σ^2\quad\text{for all }i\] then, when \(n\) is large \[\lim_{n→∞} \frac{\sum_i X_i-μ}{σ\sqrt{𝑛}} ∼ \text{Normal}(0,1)\]
If \(X_i\) is a set of i.i.d. random variables, with \[𝔼X_i=μ\quad\text{and}\quad 𝕍X_i=σ^2\quad\text{for all }i\] then, when \(n\) is large \[\lim_{n→∞} \frac{\sum_i X_i-μ}{\sqrt{𝑛}} ∼ \text{Normal}(0, σ^2)\]
≈95% of normal population is between \(-2\cdot\text{sd}(\mathbf x)\) and \(2\cdot\text{sd}(\mathbf x)\)
≈99% of normal population is between \(-3\cdot\text{sd}(\mathbf x)\) and \(3\cdot\text{sd}(\mathbf x)\)