joint distributions Flashcards
1
Q
joint distributions and marginals
A
P(X=xi, Y=yi) = p_ij; note this may be notated with P(xi,yi), and reads: probability of both events occurring
marginal distribution: eg P(X=xi) = sum_j P(X=xi,Y=yj)
2
Q
independent random variables
A
X and Y are independent if their joint probability mass function is equal to the product of their two marginal distributions
P(X=xi,Y=yj) = P(X=xi,Y=*)P(X=*,Y=yj) (for all i,j) (similarly for continuous distributions)
3
Q
covariance
A
- cov(X,Y) = E[ (X-E(X))(Y-E(Y)) ] = E(XY) - E(X)E(Y)
- independent variables have a covariance of 0
- the covariance matrix of random variables {X1,X2,…,Xn} is a symmetric matrix with each entry cov(Xi,Xj)
4
Q
correlation
A
- Pearson’s correlation
* a kind of normalized covariance
* assesses linear relationships between X and Y
* corr(X,Y) = cov(X,Y) / sqrt(var(X)var(Y)) - Spearman’s
* for X,Y, the Pearson correlation between the rank values of X and Y
* assesses monotonic relationships between X and Y
5
Q
linear combinations of random variables
A
- for Y = a1X1 + … + anXn
- E(Y) = a1E(X1)+…+anE(Xn)
- if the variables are independent:
- var(a1X1+…+anXn) = a1^2var(X1)+…+an^2var(Xn)
- if Xi are normal, then Y is also normal
- if the variables are not independent:
var(a1X1+a2X2) = a1^2var(X1)+2a1a2cov(X1,X2)+a2^2var(X2) - if the Xi are “jointly normal,” then Y is also normal, by definition
6
Q
functions (compositions) of random variables / variable transformations
A
- given random variable X, we want distribution of random variable Y, Y=g(X)
- assuming f_X is the pdf of X, this can be viewed as a rescaling of the domain of f_X
- the conversion is performed via the CDF:
- F_Y(y) = F_X(g^{-1}(y))
- ie under the new distribution, p(Y<=y) is given in terms of known functions, F_X and the inverse of g
- note statistics (eg between 2 distributions, such as p-value for difference in means) are not necessarily preserved under random variable transformations