vee · 2022年11月26日

# 为什么是二项分布

NO.PZ2017092702000083

If the probability that a portfolio outperforms its benchmark in any quarter is 0.75, the probability that the portfolio outperforms its benchmark in three or fewer quarters over the course of a year is closest to:

A.

0.26

B.

0.42

C.

0.68

C is correct.

The probability that the performance is at or below the expectation is calculated by finding F(3) = p(3) + p(2) + p(1) + p(0) using the formula:

$p{(3)}=\frac{4!}{{(4-3)}!3!}0.75^3{(1-0.75)}^{4-3}={(24/6)}{(0.42)}{(0.25)}=0.42$

$p{(2)}=\frac{4!}{{(4-2)}!2!}0.75^2{(1-0.75)}^{4-2}={(24/4)}{(0.56)}{(0.06)}=0.20$

$p{(1)}=\frac{4!}{{(4-3)}!1!}0.75^1{(1-0.75)}^{4-1}={(24/6)}{(0.75)}{(0.02)}=0.06$

$p{(0)}=\frac{4!}{{(4-0)}!0!}0.75^0{(1-0.75)}^{4-0}={(24/24)}{(1)}{(0.004)}=0.004$

Therefore

F(3) = p(3) + p(2) + p(1) + p(0)= 0.42 + 0.20 + 0.06 + 0.004= 0.684 or approximately 68 percent.

①正好有3个季度超过benchmark

②超过benchmark的季度数少于3个季度（2个，1个和0个季度）