1. Independence of random variables
Definition
Random variables X and Y are independent if their joint distribution function factors into the product of their marginal distribution functions
Theorem
Suppose X and Y are jointly continuous random variables. X and Y are independent if and only if given any two densities for X and Y their product is the joint density for the pair (X,Y) i.e.
Proof:
If X and Y are independent random variables and Z =g(X), W = h(Y) then Z, W are also independent.
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2. Example
Suppose X and Y are discrete random variables whose values are the non-negative integers and their joint probability function is
Are X and Y independent? What are their marginal distributions?
Factorization is enough for independence, but we need to be careful of constant terms for factors to be marginal probability functions.
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3. Example and Important Comment
The joint density for X, Y is given by
Are X, Y independent?
Independence requires that the set of points where the joint density is positive must be the Cartesian product of the set of points where the marginal densities are positive i.e. the set of points where fX,Y(x,y) >0
must be (possibly infinite) rectangles.
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4. Conditional densities
If X, Y jointly distributed continuous random variables, the conditional density function of Y | X is defined to be
if fX(x) > 0 and 0 otherwise.
If X, Y are independent then
Also,
Integrating both sides over x we get
This is a useful application of the law of total probability for the continuous case.
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5. Example Consider the joint density
Find the conditional density of X given Y and the conditional density of Y given X.
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6. Properties of Expectations Involving Joint Distributions
For random variables X, Y and constants
E(aX + bY) = aE(X) + bE(Y)
Proof:
For independent random variables X, Y
E(XY) = E(X)E(Y)
whenever these expectations exist.
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7. Covariance Recall: Var(X+Y) = Var(X) + Var(Y) +2 E[(X-E(X))(Y-E(Y))]
Definition
For random variables X, Y with E(X), E(Y) < ∞, the covariance of X and Y is
Covariance measures whether or not X-E(X) and Y-E(Y) have the same sign.
Claim:
Proof:
Note: If X, Y independent then E(XY) =E(X)E(Y), and Cov(X,Y) = 0.
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8. Example
Suppose X, Y are discrete random variables with probability function given by
Find Cov(X,Y). Are X,Y independent?
y x -1 1
pX(x)
1/8
pY(y)
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9. Important Facts
Independence of X, Y implies Cov(X,Y) = 0 but NOT vice versa.
If X, Y independent then Var(X+Y) = Var(X) + Var(Y).
If X, Y are NOT independent then
Var(X+Y) = Var(X) + Var(Y) + 2Cov(X,Y).
Cov(X,X) = Var(X).
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10. Example
Suppose Y ~ Binomial(n, p). Find Var(Y).
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11. Properties of Covariance
For random variables X, Y, Z and constants
Cov(aX+b, cY+d) = acCov(X,Y)
Cov(X+Y, Z) = Cov(X,Z) + Cov(Y,Z)
Cov(X,Y) = Cov(Y, X)
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12. Correlation Definition
For X, Y random variables the correlation of X and Y is
whenever V(X), V(Y) ≠ 0 and all these quantities exists.
Claim:
ρ(aX+b,cY+d) = ρ(X,Y)
Proof:
This claim means that the correlation is scale invariant.
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13. Theorem
For X, Y random variables, whenever the correlation ρ(X,Y) exists it must satisfy
-1 ≤ ρ(X,Y) ≤ 1
Proof:
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14. Interpretation of Correlation ρ
ρ(X,Y) is a measure of the strength and direction of the linear relationship between X, Y.
If X, Y have non-zero variance, then
If X, Y independent, then ρ(X,Y) = 0. Note, it is not the only time when ρ(X,Y) = 0 !!!
Y is a linearly increasing function of X if and only if ρ(X,Y) = 1.
Y is a linearly decreasing function of X if and only if ρ(X,Y) = -1.
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15. Example
Find Var(X - Y) and ρ(X,Y) if X, Y have the following joint density
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16. Conditional Expectation
For X, Y discrete random variables, the conditional expectation of Y given
X = x is
and the conditional variance of Y given X = x is
where these are defined only if the sums converges absolutely.
In general,
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17. For X, Y continuous random variables, the conditional expectation of
Y given X = x is
and the conditional variance of Y given X = x is
In general,
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18. Example
Suppose X, Y are continuous random variables with joint density function
Find E(X | Y = 2).
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19. More about Conditional Expectation
Assume that E(Y | X = x) exists for every x in the range of X. Then,
E(Y | X ) is a random variable. The expectation of this random variable is
E [E(Y | X )]
Theorem
E [E(Y | X )] = E(Y)
This is called the “Law of Total Expectation”.
Proof:
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20. Example
Suppose we roll a fair die; whatever number comes up we toss a coin that many times. What is the expected number of heads?
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21. Theorem For random variables X, Y V(Y) = V [E(Y|X)] + E[V(Y|X)] Proof:
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22. Example Let X ~ Geometric(p).
Given X = x, let Y have conditionally the Binomial(x, p) distribution.
Scenario: doing Bernoulli trails with success probability p until 1st success so X : number of trails. Then do x more trails and count the number of success which is Y.
Find, E(Y), V(Y).
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23. Law of Large Numbers Toss a coin n times. Suppose
Xi’s are Bernoulli random variables with p = ½ and E(Xi) = ½.
The proportion of heads is
Intuitively approaches ½ as n  ∞ .
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24. Markov’s Inequality
If X is a non-negative random variable with E(X) < ∞ and a >0 then,
Proof:
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25. Chebyshev’s Inequality
For a random variable X with E(X) < ∞ and V(X) < ∞, for any a >0
Proof:
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26. Back to the Law of Large Numbers
Interested in sequence of random variables X1, X2, X3,… such that the random variables are independent and identically distributed (i.i.d).
Let
Suppose E(Xi) = μ , V(Xi) = σ2, then
and
Intuitively, as n  ∞, so
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27. Formally, the Weak Law of Large Numbers (WLLN) states the following:
Suppose X1, X2, X3,…are i.i.d with E(Xi) = μ < ∞ , V(Xi) = σ2 < ∞, then for any positive number a
as n  ∞ .
This is called Convergence in Probability.
Proof:
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28. Example Flip a coin 10,000 times. Let E(Xi) = ½ and V(Xi) = ¼ .
Take a = 0.01, then by Chebyshev’s Inequality
Chebyshev Inequality gives a very weak upper bound.
Chebyshev Inequality works regardless of the distribution of the Xi’s.
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29. Strong Law of Large Number
Suppose X1, X2, X3,…are i.i.d with E(Xi) = μ < ∞ , then converges to μ
as n  ∞ with probability 1. That is
This is called convergence almost surely.
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