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Sampling, combinations and CLT
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Understanding how the sums of random variables behave, and how sums of large samples of any random variable is roughly normally distributed.
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Linear Transformation of Random Variables
AHL AI 4.14
Suppose that the random variable X is scaled and shifted, producing the random variable aX+b. The expected value and variance of the resulting variable are
E(aX+b)Var(aX+b)=aE(X)+b=a2Var(X)
Expected value of linear combination of random variables
AHL AI 4.14
For random variables X and Y, the expected value of a linear combination of X and Y is equivalent to the sum of the transformed variables' expectations. That is,
E(aX+bY+c)=aE(X)+bE(Y)+c
We can extend this result to the linear combination of any number of independent random variables X1,…Xn:
E(a1X1+a2X2+⋯+anXn+c)
=a1E(X1)+a2E(X2)+⋯+anE(Xn)+c
This rule is sometimes referred to as the linearity of expectation.
Variance of linear combination of random variables
AHL AI 4.14
If X and Y are independent random variables, the variance of a linear combination of X's and Y's is equivalent to the sum of the transformed variances. That is,
Var(aX+bY+c)=a2Var(X)+b2Var(Y)
We can extend this result to the linear combination of any number of independent random variables X1,…Xn:
Var(a1X1+a2X2+⋯+anXn+c)
=a12⋅Var(X1)+a22⋅Var(X2)+⋯+an2⋅Var(Xn)
Notice that the variance of the random variable that represents two different observations from one population, Var(X1+X2)=Var(X1)+Var(X2), is not equal to the variance of one observation doubled, Var(2X1)=4Var(X1).
Sample mean expected value and variance
AHL AI 4.15
Suppose X1,X2,…Xn are n independent observations of the random variable X. We can find the mean of this sample by adding together the observations and dividing by n. This defines a new random variable, Xˉ, called the sample mean of the Xi:
Xˉ=nX1+X2+⋯+Xn
Since Xˉ is a linear combination of independent random variables, its expected value and variance are given by
E(Xˉ)Var(Xˉ)=E(X)=nVar(X)
Linear combination of normal random variables
AHL AI 4.15
Any linear combination of normally distributed random variables follows a normal distribution.
For independent random variables X and Y with X∼N(μX,σX2), Y∼N(μY,σY2), then the random variable W given by W=aX+bY+c follows the distribution
W∼N(aμX+bμY+c,a2σX2+b2σY2)
We can extend this result to the linear combination of any number of independent, normally distributed random variables.
Central Limit Theorem
AHL AI 4.15
The Central Limit Theorem states that if X1,X2,…Xn are independent observations of the random variable X, with E(Xi)=μ and Var(Xi)=σ2 for any i∈{1,2,…n}, then for sufficiently large n (typically n>30), regardless of the distribution of X, the sum of the Xi's is normally distributed:
i=1∑nXn∼N(nμ,nσ2)
Further, the random variable Xˉ=nX1+X2+⋯+Xn is normally distributed with mean μ and variance nσ2:
Xˉ∼N(μ,nσ2)
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