By Aliakbar Montazer Haghighi

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2) we have: P {X ≥ 1200} ≤ E ( X ) 600 1 = = . 3) we have: P { X − 600 ≥ 120} ≤ σ2 1 = . (120 )2 120 52 Probability and StatiSticS Therefore, P { X − 600 < 120} ≥ 1 − 1 119 = . 4. The Weak Law of Large Numbers Let {X1, X2, . . , Xn, . . } be a sequence of independent and identically distributed (iid) random variables with mean μ. Next, for an arbitrary small positive number ε, we have: lim P n →∞ { X1 + X 2 + n + Xn } − µ > ε = 0. 11) Proof: Since {X1, X2, . . , Xn, . } is iid, due to additivity property of expected value, we have: E + X n nµ = µ.
Hence, E (X ) ≥ ∞ ∑ xf ( x). 5) we have: E (X ) ≥ ∞ ∞ x=a x=a ∑ af ( x) = a∑ f ( x). 6) functionS of a random Variable 51 Hence, E ( X ) ≥ aP {X ≥ a} . 3) follows. 3. Chebyshev’s Inequality Let X be a nonnegative random variable with finite mean μ and variance σ2. Let k also be a fixed positive number. Hence: P { X − µ > k} ≤ σ2 . 8) Proof: Consider the random variable (X − μ)2, which is positive. 2), we now have: { } P (X − µ) ≥ k2 ≤ 2 ) 2 E ( X − µ ) . 9) Now (X − μ)2 ≥ k2 implies that X − μ ≥ k.
Xr then, is similarly defined as: Px1, x2,…, xr = P ( X 1 = x1, X 2 = x2 , … , X r = xr ) . Notes: (1) From the axioms of probability, a joint probability mass function has the following properties: random Vector 43 (a) px1 x2, , xr ≥ 0, and (b) ∑ x1 ∑ x2 ∑ xr px1x2 xr = 1. (2) The discrete bivariate, pxi, yj means the probability that X = xi, Y = yj, and P is defined on the set of ordered pairs {(xi, yj), j ≤ i ≤ m, i ≤ j ≤ n} by pxi , yj = P ([ X = xi ] and [Y = y j ]). We may obtain each individual distribution functions from the joint distribution.