**Mumbai University > Electronics and Telecommunication Engineering > Sem 5 > Random Signal Analysis

Marks: 10M

Year: May 2016

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 ${X_n }^{ ̅ }$=1($X_1$+$X_2$+⋯.$X_n$)/n

then for any ∈>0P $lim_{n→∞}$ P(|${X_n }^ ̅ $-μ|>∈)=0 ..................... (1)

This is known as weak law of large numbers and $(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 ($X_n$) ̅=1($X_1$+$X_2$+⋯.$X_n$)/n

then for any ∈>0P ($lim_{n→∞}$ |($X_n$ ) ̅ -μ|>∈)0........................ (2)

This is known as weak law of large numbers and ($X_n$ ̅ 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.