10%(b) What is the distribution of X?

Problem 12. (2.5 %) Let Xi, 1 ≤ i ≤ 4 be independent Bernoulli random variable each with mean p = 0.1. Let X = That is, X is a Binomial random variable with parameters n = 4 and p = 0.1. Then, (i) E[X1|X=2]=0.1(ii) E[X1| X=2]=0.5(iii) EXi|X=2]=0.25

Problem 11. (2.5 %) To obtain a driving license, John needs to pass his driving test. Every time John takes a driving test, with probability 1/2, he will clear the test independent of his past. John failed his first test. Given this, let Y be the additional number of tests John takes before obtaining a license. Then, (i) E[Y] =1 (ii) E[Y] =2 (iii) E[Y]=0

Problem 13. (2.5 %) Let X1, X2, X3 be independent random variables with the continuous distribution over [0,1]. Then P(X1＜X2＜X3) = (i) 1/6 (ii) 1/3 (iii) 1/2 (iv) 1/4

(a) (3%) Find , where the condition B ={ X ≥μx}

(b) (3%) What is E[X|B]?

10%(a) Find the moment generating function of X.

(12%) Let V be a finite-dimensional vector space and T :V →V be linear. Suppose rank(T) = rank(T2). Prove that R(T) ∩N(T) = {0}.

(13%) Given a vector space V over F. Define the dual space of V* of V as the set of all functions (also known as linear functionals) from V to F, i.e, V* {f|f : V →F}. It is obvious that V* is itself also a vector space with the addition + :V*'x V*→ V* and scalar multiplication * : F x V*→  V* defined as pointwise addition as well as pointwise scalar multiplication. Given any linear transformation T : V→  W. The transpose Tt is a linear transformation from W* to V* defined by Tt(f) = fT for any f W*. For every subset S of V, we define the annihilator Suppose V, W are both finite-dimensional vector spaces and T : V → W is linear. Prove that .N(Tt) = (R(T))0.

(10%) The nullities of the matrices BBT - λ/ for  λ= 0, 1,2,3, 4  are______,________,_________,_______ respectively. 

(d) (2%) Compute dim(N(Bt A)).

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