2 万美元投资分配

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英文解析

A "stars and bars" approach is the easiest. $^{29}$ Let each star denote a thousand- dollar investment and each bar denote a line of demarcation between funds.

Then *****|************ has 20 stars and four bars to indicate an allocation of $(\mathbb{S}5,000;\mathbb{S}8,000;\mathbb{S}0;\mathbb{S}7,000;\mathbb{S}0)$ . There are $\left( \begin{array}{c}{24}\\ {4} \end{array} \right) = \frac{24!}{4!(24 - 4)!} =$ $\begin{array}{r}\frac{24\cdot 23\cdot 22\cdot 21}{4\cdot 3\cdot 2\cdot 1} = 10,626 \end{array}$ ways to allocate the four bars to the 24 possible locations for the 24 symbols I used. The general answer is $\left( \begin{array}{c}{N + k - 1}\\ {k - 1} \end{array} \right)$ for $N$ - thousand dollars invested into $k$ funds. Please confirm that in the case $N = 20,k = 2$ this formula gives the simple answer $N + 1 = 21$

A less elegant approach is to consider 20 independent generalized Bernoulli trials where we roll a five- sided die and record each result as a five- tuple with a 1 in the position corresponding to the die's outcome and zeroes otherwise. The five- tuple containing the cumulative total count of outcomes is distributed multinomial (DeGroot [1989, p. 297]). The number of possible fungible (i.e., fi* nancially indistinguishable) allocations of bils to buckets must be the same as the size of the sample space in this generalized experiment. For a multinomial with $N$ generalized Bernoulli trials (each of $k$ outcomes) the sample space is of size $\left( \begin{array}{c}{N + k - 1}\\ {k - 1} \end{array} \right)$ as above. 20

Story: 1 spoke with a gay employee at a big- name Wall Street investment bank. I told him that some of my students were about to fly out to New York to interview. He told me: "Make sure they have nice suits, good hair- cuts, and wear their wedding rings."