(一)求機率密度函數中的常數C。(5分)

(1) (5%) Find all solutions to the system Ax = b.

(2) (5%) Find the bases for the column space of A and the null space of A. 

Problem 22. (3.0 %) Suppose you play a matching coin game with your friend as follows. Both you and your friend have a coin. Fach time, you two reveal a side (i.e. H or T) of your coin to each other simultaneously. If the sides match, you WIN a 1 from your friend if sides do not match then you lose a 1 your friend. You decide to go with the following strategy. Every time, you will toss your unbiased coin independently of everything else, and you will reveal its outcome to your friend (your friend does not know the outcome of your random toss until you reveal it). Then, (i) On average, you will lose money to your smart friend (ii) On average, you will lose neither lose nor win. That is, your average gain/loss is 0. (iii) On average, you will make money from your friend

(二)求P(X＜4)(5分)

(1) (10%) Find conditional probability mass function of X given the event B.

(2) (10%) Find the conditional variance of X given the event B.

(10%) Let X be a random variable with the following probability mass function (PMF): Let Y = . Find the cumulative distribution function (CDF) of Y.

a) (5%) Find the pdf of Y = aX+ b, where a ≠ 0 and b are real numbers.

b) (5%) Find the pdf of Z = |X].

a) (5%) Find the pdf of Y = aX + b, where a≠ 0 and b are real numbers.