1.Using the summation operator method to find sigma i^5 (for all i=1 to n)
2.Find the coefficient of x^24 in each of the following:
(a) (1+x+x^2+x^3+......+x^6)^5
(b) (1+2x)^2/(1-2x)^4
(c) (x^3+x)/(1-2x)^3
(d) 1/(1+x)(1-3x)(2x+3)
(e) (1-2x)^(-4/3)
3.Prove that the number of partitions of r into parts each of which appears
at most twice is equal to the number of partitions of r into parts the
sizes of which are not divisible by 3
4.How many 20-digit ternary (0,1,2) sequences are there where:
(a) There is at least on 2 and odd number of 0's?
(b) No symbol occurs exactly twice?
(c) No symbol occurs exactly three times?
(d) There are exactly two 2's or none at all?
5.Suppose that X is a discrete random variable with probability
distribution given by Pr(X=x)={ k(1/4)^x, x=0,1,2.....}
0, otherwise
where k is a constant, determine
(a) k=? (b)母體平均? 標準差?
6.What probability can we select six nonconsecutive integers from {1,2,3,..,37}
7.If Y is geometric random variable with E(Y)=7/3, determine
(a) Pr(Y=3)
(b) Pr(Y>=3)
(c) Pr(Y>=5)
(d) Pr(Y>=5 | Y>=3)
(e) Pr(Y>=6 | Y>=4)
(f) Var(Y)
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