Lecture 5 Flashcards
Define the characteristic function of X and state why and how it’s useful for our purposes.
Define convergence in distribution.
How many discontuinity points or jumps can a limiting distribution have?
If X_i iid with given mean and sigma squared, what can you say about the limiting distribution?
Consider X_i, a sequence of U[0,\theta] RVs. Give the distribution of an estimator for \theta.
State the relationship or lack thereof between convergence in distirbution and in r-th mean.
State and show by example the relationship between convergence in distribution and r-th mean.
Notice the we can realize the rv only takes values 0 and n from the discontinuity point and the rest follows.
Is there any case when convergence in distribution does imply convergence in r-th mean? What is it?
State two necessary and sufficient conditions for convergence in distribution.
State the theorem relating convergence in distribution of a vector and the convergence in distribution of it’s linear combination.
State and prove the theorem relating convergence in distribution of a vector and the convergence in distribution of it’s linear combination.
Give the characteristic function of a normal RV.
Give the definition of a k-variate normal.
State the theorem defining the distribution of a linear combination of normal RVs.
Consider we have a vector where each element is a normal RV. Does this imply that the vector is a normal RV?
State and prove the theorem defining the distribution of a linear combination of normal RVs.
Consider a scalar random variable. State the two relationships between convergence in distribution and convergence in probability.
State and prove the relationship between convergence in distribution and convergence in probability for a scalar random variable, when there is not necessarily convergence to a constant.
State and prove the relationship between convergence in distribution and convergence in probability for a scalar random variable, when there is convergence to a constant.
State the two relationships between convergence in distribution and convergence in probability for vectors.
State and prove the relationship between convergence in probability and convergence in distribution for vectors when there is no convergence to a constant.
State and prove the relationship between convergence in probability and convergence in distribution for vectors when there is convergence to a constant.
