EC124 T2: Univariate and Bivariate distributions Flashcards
(16 cards)
How do we define a discrete random variable with a probability density function (4,1)
-Function X assigns to each element of C a real number X(c) = x
-Suppose X is a scalar random variable with a finite number of values
-Let Px(x) be a function where px(x) > 0
-∑Px(X) = 1
-Then X is a discrete random variable with probability density function PC(x), and p(a<x<b) = ∑b, x = a f(x)
What is the cumulative distribution function of a discrete random variable (3)
-fx(x0) = p(x≤0) = ∑x≤0 px(c)
-This is monotonic, as more is always better
-p(a<x<b) = f(b) - f(a)
How do you work out the median, mode and mean/expectation for a discrete random variable (2,1,2)
-Median: for discrete random variable x, if p(x≤x0) = 0.5, and p(x≥x0) = 0.5, x0 is the median
-Integrate the probability distribution function and equal it to a half
-Mode: the value of x where px(x) is maximised
-Mean/Expected value: x1P(x1) + x2P(x2) + …
-E(X) = ∑PiXi μx
How does the expected value change to transformations (2)
-E(x+a) = E(X) + a
-E(ax) = aE(X)
What is the variance and how does it change to transformations (2,2)
-The variance is the average squared deviations from the mean
-V(x) = E(X2) - (E(X))2
-V(X+a) = v(x)
v(ax) = a2v(X)
What must a valid bivariate probability density function define (2)
A probability for each pair of events (x1i, x2j) such that:
-P(x1i n x2j ≥ 0)
-ΣiΣjp(x1i n x2j) = 1
What is the marginal probability density function (2,1)
-For the bivariate case, consider event x1 = x1i, where x1 = x1i, and x2 takes any possible value
-The probability is ΣjP(x1i, x2j)
-You keep event x1 constant at one specific result, whilst changing x2
What is covariance (2,2)
-Covariance is a measure of association between 2 random variables
-Covariance measures both series in terms of deviations from their respective means and then measures the expected value of the cross-product of the 2 terms
The 2 elements of covariance include:
-A measure of sign showing +ive/-ive deviations, associated with +ive/-ive deviations of x2
-Size of how big the deviations are (bigger dev = bigger cov)
What is the formula for covariance (2)
-E(XY) - E(X)E(Y)
-E(X - E(X))(Y - E(Y))
How to graphically represent covariance (5,3)
-Have a graph with x1 on the x axis, x2 on the y axis
-On the x1 and x2 axis label all the possible events (x1 = 1, 2, 3, x2 = 1, 2)
-Draw points at each of these events (x1 = 1 n x2 = 1 …), the bigger the point representing the higher the probability of occuring
-Draw straight lines on the axis from E(X1) and E(X2)
-Q1 = Top left, Q2 = top right, Q3 = bottom left, Q4 = bottom right
-Q2 and Q3 are positive, as both (X1 - E(X1)) and (X2 - E(X2)) are either positive or negative, Q1 and Q4 are negative for similar reason
-The covariance formula is saying on average, do points lie predominantly in the +ive or -ive quadrants
What is the expected value formula for a bivariate distribution with transformations
(2)
-E(X + Y) = E(X) + E(Y)
-E(aX + bY) = aE(X) + bE(Y)
What is the variance formula for a bivariate/trivariate distribution (2)
-V(X + Y) = V(X) + V(Y) + 2cov(X, Y)
(to work out, do (x+y)2 then x2 = v(x), y2 = v(y), 2xy = 2cov(x,y))
-V(X + Y - Z) = V(X) + V(Y) + V(Z) + 2cov(X, Y) - 2cov(X, Z) - 2cov(Y, Z)
What is the conditional probability density function for X and Y (2)
-P(x1i|x2j) = p(x1i n x2j)/p(x2j)
-P(X = 7|Y=1) = P(X = 7 n Y = 1)/P(Y = 1)
What is the variance with conditionality (2)
-V(X) = E(V(X|Y)) + V(E(X|Y))
-Variance = average of conditional variances, weighted by likelihood + variance of conditional means
How can we define probability density function fx(x) for a random continuous variable (4)
-If x is a scalar random variable along the real line
-fx(x) ≥ 0
-∫fx(x) dx = 1
-Then x is a continuous random variable with probability density function fx(x)
How do you calculate the median and expected value for a continuous random variable (2)
-Median = ∫0 a f(x) = 1/2
-E(x) = ∫xf(x) dx (for all x values in the domain) = ∑xp(x)
