Question

According to the Normal model ​N(0.071​,0.031​) describing mutual fund returns in the 1st quarter of​ 2013, determine what percentage of this group of funds you would expect to have the following returns. Complete parts​ (a) through​ (d) below. ​a) Over​ 6.8%? ​b) Between​ 0% and​ 7.6%? ​c) More than​ 1%? ​d) Less than​ 0%?

Answer 1

Let X be the random variable denoting the mutual funds returns in the 1st quarter of 2013.

Thus, X ~ N(0.071, 0.031)

i.e. (X - 0.071)/0.031 ~ N(0,1)

(a) The required probability = P(X > 6.8%) = P(X > 0.068) = 1 - P(X < 0.068) = 1 - P[(X - 0.071)/0.031 < (0.068 - 0.071)/0.031] = 1 - P[(X - 0.071)/0.031 < - 0.0967] = 1 - (-0.0967) = 1 - 0.4615 = 0.5385 (Ans).

[(.) is the cdf of N (0,1)].

(b) The required probability = P(0 < X < 7.6%) = P(0 < X < 0.076) = P[(0 - 0.071)/0.031 < (X - 0.071)/0.031 < (0.076 - 0.071)/0.031] = P[-2.2903 < (X - 0.071)/0.031 < 0.1613] = (0.1613) - (-2.2903) = 0.5641 - 0.0110 = 0.5531 (Ans).

(c) The required probability = P(X > 1%) = 1 - P(X < 0.01) = 1 - P[(X - 0.071)/0.031 < (0.01 - 0.071)/0.031] = 1 - P[(X - 0.071)/0.031 < - 1.9677] = 1 - (-1.9677) = 1 - 0.0246 = 0.9754 (Ans).

(d) The required probability = P(X < 0%) = P(X < 0) = P[(X - 0.071)/0.031 < (0 - 0.071)/0.031] = P[(X - 0.071)/0.031 < - 2.2903] = (-2.2903) = 0.0110 (Ans).