SU2 - The Properties of Random Variables, The Normal and Its Related Distributions Flashcards
What is expected value of a random variable?
It is the weighted average of all possible values of X aka mean
What is the Expectation of Sums?
E(X1+X2)
E(X1)+E(X2)
What is the result of E(c1X1+c2X2)?
E(c1X1) + E(c2X2) = c1E(X1) + c2E(X2)
What is the expectation of a binomial distribution?
np
What is Jensen’s inequality?
g(E(X)) ≤ E(g(X))
What is the result of E (aX + b)?
E (aX + b) = aE(X) + b
What is the expectation of a Bernoulli distribution?
p
Is the expectation of the function of a random variable equals to the function of the random variable’s expectation?
E [g (X)] = g (E [X])
No, unless g(X) is a linear function, Jensen’s inequality
Is the expectation of the product of random variables equal to the product of their expectations?
No, only if the random variables with the product are independent
What is the Cauchy distribution?
Student t distribution with 1 degree of freedom. In this case, the expectation may not exist
What is the variance?
To measure how “spread out” the values of a random variable are
𝑉𝑎𝑟𝑋=𝐸[(𝑋−𝜇)2] or 𝐸(𝑋2)−𝜇2
For any constant c, what is the value of Var(c)?
- For any constant, value of variance is 0 because there is no variance at all.
For any constants a and b, Var(aX+b) = ??
For any constants a and b, Var(aX+b) = a^2 Var(X)
When we add b to X, we only shift the distribution of X laterally. The shape of the distribution of X stays the same; therefore, the variance of X remains unchanged
For any constants a and b, and sd ( aX+b ) = ??
|a| sd (X)
What does standardization mean?
To re-centre the expectation of the random variable to 0 and to normalise its variance to 1, i.e. E(X) = 0 and Var(Z) = 1.
If 𝑋1 and 𝑋2 are uncorrelated/independent, Var𝑋1+𝑋2 =?
Var(X1) + Var(X2)
Is the variance of the sum of random variables always equal to the sum of their variances?
No, unless the variables are independent
What happens to the covariance when X and Y are independent?
If X and Y are independent, then Cov(X,Y) = 0, and E(XY) = E(X)E(Y)
What is the correlation coefficient?
The correlation coefficient can be thought of as a standardised covariance that does not depend on units of measurement.
corr(X,Y) = 1(-1) means a perfect positive (negative) linear relationship
What is the variance of a bernoulli distribution?
Var(X) = np(1-p)
What is the alternate expression for Cov(X, Y)?
E(XY) - 𝜇𝑋𝜇𝑌
E(XY) - E(X)E(Y)
What are the three properties of covariances?
- For any constant a, Cov(a,X) = 0.
- For random variable Z, Cov(X+Y,Z) = Cov(X,Z)+Cov(Y,Z).
- For any constants a1 and a2, Cov(a1 X,a2 Y) = a1 a2 Cov(X,Y).
Does zero correlation imply independence?
Yes
What happens if you add a constant to a random variable in a covariance?
Does not affect its covariance with another random variable
E(X) = E(E(X│Y)) = E(c) = c
What happens if you multiply a constant to a random variable in a covariance?
Affect the covariance by the same multiple
What is a conditional expectation?
Explaining one variable in terms of the other variable
What is the law of Iterated Expectation?
E[E(Y|X)] = E(Y)
How to write a conditional expectation of a random variable Y given a random variable X?
E(Y|X)
E(X|X) = ?
X
E(c(X)|X) = ?
c(X)
if X and Y are independent, E(Y|X) = ?
E(Y)
if E(Y|X) = E(Y), then Cov(X,Y) = ?
0
E(XY) can directly jump to E(E(XY|Y)?
Yes
What is a normal random variable?
The normal random variable is symmetrically distributed around its mean μ. The peak of the normal density function occurs at Y=μ and the density function is concentrated around μ.
How to read this?
X ∼ N(µ, σ2)
The normal distribution with mean µ and variance σ2
If X ∼ N(µ, σ2) and if Y = aX+b and a =/= 0, then y = ?
Y has a normal distribution with mean aµ+b and variance a2o2
every linear function of X will also have a normal distribution.
What is the standard normal?
A normal random variable with mean 0 and variance 1.
What is the Chi-Square mean and variance?
E(Y) = m Var(Y) = 2m
How is the chi-square related to the standard normal?
When the degrees of freedom in a Student-t goes to infinity, the Student-t becomes the standard normal.
