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Explain strong law of large numbers and weak law of large numbers

Mumbai University > Electronics and Telecommunication > Sem5 > Random Signal Analysis

Marks: 5M

Year: May 2015

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Weak Law of Large numbers:

Let $X_1,X_2,X_3…..X_n$ be a sequence of independent identically distributed RVs each with finite mean E$(X_i )$=μ

Let $(\bar{X_n} ) =\frac1{n}(X_1+X_2+⋯.X_n)$

then for any ∈>0 $\lim_{n\to∞}⁡P(|(\bar{X_n} ) -μ|\gt∈)=0$ ------ (1)

This is known as weak law of large numbers and $\bar{X_n}$ is sample mean

Strong Law of Large numbers:

Let $X_1,X_2,X_3…..X_n$ be a sequence of independent identically distributed RVs each with finite mean

E($X_i$ )=μ

Let $(\bar{X_n} ) =\frac{1}n(X_1+X_2+⋯.X_n)$

then for any ∈>0 $P(\lim_{n\to∞} |(\bar{X_n} ) -μ|\gt∈)=0$ ------ (2)

This is known as weak law of large numbers and $(\bar{X_n} )$ is sample space

Here, in equation (1) we take limit of probabilities and it tells us how sequence of probability converges

In equation(2) we take the probability of the limit and tells us how the sequence of random variables behave in the limits.