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If you make n independent trials with the same random variable X, which has expected value μ and standard deviation σ, you can then sum the random variables. That sum S will be approximately normally distributed when n is sufficiently large.

Rule

The Central Limit Theorem

S=X1+X2++Xn

has expected value

E(S)=nμ

and standard deviation

SD(S)=nσ

Example 1

You let Xi be a randomly selected cheese at the cheese counter. The expected value of the weight of such a cheese is μ=1.2kg, with a standard deviation of σ=0.3kg. If you let S denote the sum of 50 such cheeses, what is the expectation value and the standard deviation for S?

You know that

E(S)=E(X1)+E(X2)++E(X50)=nμ

E(S)=E(X1)+E(X2)++E(X50),=nμ

and that the expected value is

E(S)=501.2=60

You also know that

Var(S)=Var(X1)+Var(X2)++Var(X50)=nσ2

Var(S)=Var(X1)+Var(X2)++Var(X50)=nσ2

which gives the standard deviation

SD(S)=nσ=500.32.12

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