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Lecture11

# Lecture11 - Example Suppose we have particles emitted in t...

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•1 Example: Suppose, we have particles emitted in t seconds according to a Poisson law with rate Assume also that all particles have energies that are distributed according to the Maxwell distribution with mean and variance Let S denote the total energy emitted in t seconds. Clearly, it is a random variable. Let us , for instance, compute its mean and variance. We have . t λ }, { k X 2 3 T . 2 3 2 T t N t EN = = ) var( , Then . 2 3 t T EX EN ES = = 2 ) )( var( ) var( ) var( EX N X EN S + = 4 15 4 9 2 3 2 2 2 t T T t T t = + = ± Weak Law of Large Numbers Here we begin to study the behavior of random variables and as n goes to infinity. It is important to estimate unknown parameters of the distribution from which you take samples. The weak law of large numbers(WLLN) says that for sufficiently large n random variables: with a high probability takes values close to the mean of the distribution from which you sample. n S n M n M Theorem 3 with finite mean m and variance . Then for any fixed Proof: Applying Chebyshev’s inequality, we obtain Therefore, as +∞ = 1 } { i i X 2 σ 0 > ε 1 ) | (| lim = < +∞ m M P n n 2 2 ) | (| n m M P n 1 1 ) | (| 2 2 < n m M P n . +∞ n

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•2 Example: Let be an IID-sequence with EX=m and Let Does the WLLN hold for since we know that the WLLN holds for +∞ } { i X . ) var( 2 σ = X . , 3 2 N n X Y n n + = ? } { 1 +∞ = i n Y {} 1 2 1 3 2 3 2 1 1 1 1 1 = +∞ = +∞ = +∞ = < = = < + = = < n i i n n i i n n i i n m X n P m X n P Y E Y n P ) | (| lim ) | (| lim ) | (| lim ε +∞ = 1 } { i i X ± Strong Law of Large Numbers We know that is an IID– sequence , then the WLLN holds. Here we point out that an even stronger result takes place. Theorem 4 Consider a sequence of IID random variables . Let E{X}=m and . Then +∞ = 1 } { i i X +∞ = 1 } { i i X 2 ) var( = X . 1 ) lim ( = = m M P n n ± Practical Significance of SLLN The SLLN is the basis for estimating different quantities in real life.
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Lecture11 - Example Suppose we have particles emitted in t...

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