On One Kind Of Coupon Collection Problems

The coupon collection problem has a long history. This dissertation will mainly discuss the relationship among the time we need to collect all the coupons, the amounts of every coupon we need to collect and the probability of every coupon comes up, at the condition of the total amount of the coupon we need to collect is fixed.In chapter Two, we will discuss the problem of two kinds of coupons collection. In this chapter, we will suppose the total amount of the coupon we need to collect is n, the time we need to collect all of the coupon is T. We will also suppose the first coupon we need to collect is n1 and the probability it comes up is p1; the second coupon we need to collect is n2 and the probability it comes up is p2.Theorem 2.1.1 and Theorem 2.1.2 will discuss the random sequence of T, Theorem 2.1.1 will proof at the condition of p1=p2, if n1, increase from 0 to n, T will decrease at first, and then increase. What's more, whatever n is even or odd, we will find out the value of n1, when T reach the minimum value respectively. Theorem 2.1.2 will proof at the condition of p1>p2, and n1≤n-1/2, if ft, increase from 0, T will decrease. And respectively, at the condition of p1< p1 and n1≥n-1/2, if n1 increase, T will increase. Theorem 2.2.2, Theorem 2.2.3 and Theorem 2.2.4 will discuss the properties of E[T]. Theorem 2.2.2 will proof at the condition of p1=p2, if n1 increase from 0 to n, E[T] will decrease at first, and then increase. What's more, whatever n is even or odd, we will find out the value of n, when T reach the minimum value respectively. Theorem 2.2.3 will proof at the condition of p1≠p1,if n1, equals the p1 quantile of b(n,p1). E[T] will reach the minimum value. At the same time, we will proof at the condition of n1 smaller than the p1 quantile of b(n,p1), if p1 increase from 0, E[T] will decrease; at the condition of n1, bigger than the p1 quantile of b(n,p1), if p1 increase, E[T] will increase. Theorem 2.2.4 will proof at the condition of n1,n2, is fixed, we can find a p1, if p1<p1 and increase,p1 will decrease; if p1>p1 and increase, p1 will increase. At the same time, we can proof E[T] will reach the minimum value if p1=p1, which is n1/P12B(n1;n,p1).In Chapter Three, we will discuss the problem of three kinds of coupons collection. At first, we will compute E[T] at the condition of three kinds of coupons. Then, we will display an example, use Mathematica to draw several pictures and discuss this example in detail. At last, we can obtain the conclusion which is close to reality.