16-Variance
I Random Variables: Variance and Covariance¶
I.1 Variance（方差）¶
[!QUOTE]
In probability theory and statistics, the variance is a way to measure how far a set of numbers is spread out. Variance describes how much a random variable differs from its expected value. The variance is defined as the average of the squares of the differences between the individual (observed) and the expected value.
[!DEFINITION 16.1]
(Variance). For a r.v. X with expectation E[X] = µ, the variance of X is defined to be \(Var(X) = E[(X − µ)^{2}]\) . The square root \(σ(X) := \sqrt{ Var(X) }\) is called the standard deviation of X.
[!THEOREM 15.3]
For a r.v. X with expectation E[X] = µ, we have \(Var(X) = E[X^{2}]-\mu^{2}\) . and \(Var[cX] = c^{2}Var[X]\) .
and we can even get that:
I.2 Covariance¶
[!DEFINITION 16.2]
(Covariance). The covariance of random variables X and Y , denoted Cov(X,Y), is defined as
\[Cov(X,Y) = E[(X − \mu_{X} )(Y − \mu_{Y} )] = E[XY]−\mu_{X}\mu_{Y}\]where µX = E[X] and µY = E[Y].
here are some important facts about covariance:
An example for the converse is not true is given in Q 1 (a).
[!DEFINITION 16.3]
(Correlation). Suppose X and Y are random variables with σ(X) > 0 and σ(Y) > 0. Then, the correlation of X and Y is defined as
\[Corr(X, Y) = \frac{Cov(X, Y)}{\sigma(X)\sigma(Y)} \in [-1, 1]\]
看到这里，其实已经可以回忆起高中学习的线性规划了。
II Practice¶
Q 1 Double-Check Your Intuition Again
(a) You roll a fair six-sided die and record the result X. You roll the die again and record the result Y.
(i) What is cov(X +Y,X −Y)?
(ii) Prove that X +Y and X −Y are not independent.
[!INFO]
协方差的双线性
cov(aX+bY,cZ)=ac⋅cov(X,Z)+bc⋅cov(Y,Z)
cov(X+Y,cZ)=c⋅cov(X,Z)+c⋅cov(Y,Z)
对于 (i)
对于 (ii) ，X+Y 和 X-Y 肯定有关系，我们举一个反例即可：
例如 \(P[X+Y = 5, X-Y = 0] = 0\quad != P[X+Y=5]* P[X-Y=0]\)
For each of the problems below, if you think the answer is "yes" then provide a proof. If you think the answer is "no", then provide a counterexample.
(b) If X is a random variable and Var(X) = 0, then must X be a constant?
yes, just need to know how to calculate Var(X).
(c) If X is a random variable and c is a constant, then is Var(cX) = cVar(X)?
no
(d) If A and B are random variables with nonzero standard deviations and Corr(A,B) = 0, then are A and B independent?
no, just see (a)
(e) If X and Y are not necessarily independent random variables, but Corr(X,Y) = 0, and X and Y have nonzero standard deviations, then is Var(X +Y) = Var(X) +Var(Y)? The two subparts below are optional and will not be graded but are recommended for practice.
(f) If X and Y are random variables then is E[max(X,Y)min(X,Y)] = E[XY]?
yes, it is obvious since max(X,Y)min(X,Y) = XY is always true.
(g) If X and Y are independent random variables with nonzero standard deviations, then is Corr(max(X,Y),min(X,Y)) = Corr(X,Y)?
It is difficult.
