Percentage daily returns on a financial asset is modelled through a normal random variable. You want to classify returns that differ from the mean by more than $3.4$ standard deviation as "abnormal" returns and returns that differ from the mean by less than $0.72$ standard deviation as "weak" returns.
a) The probability of observing an "abnormal" return is?
b) The probability of observing a "weak" return is?
Hi,
For the above question I found the answer as $0.10\%$ and $16.98\%$ but they turned out to be $0.07\%$ and $52.85\%$ respectively.
What am I doing wrong? You can find the standard normal distribution table in the link below,
$0.0003*3.4 = 0.00102 = 0.10\%$ ($0.0003$ is the value in the table),
$0.2358*0.72 = 0.169776 = 16.98\%$ ($0.2358$ is the value in the table)
You should not be multiplying by $3.4$ or by $0.72$, though multiplication by $2$ is meaningful:
The numbers you should be taking from the table are $P(Z \le -3.4) =0.00034$ and $P(Z \le -0.72) =0.23576$ from the first table and perhaps $P(Z \le +3.4) =0.99966$ and $P(Z \le +0.72) =0.76424$ from the second table
You then get
$P(|Z|\ge 3.4) = P(Z \le -3.4) + P(Z \ge +3.4) $ $= 0.00034 +(1-0.99966) $ $=2\times0.00034$ $ =0.00068$ or about $0.07\%$
$P(|Z|\le 0.72) = P(Z \le +0.72) - P(Z \le -0.72) $ $= 0.76424 -0.23576 $ $=1-2\times0.23576$ $ =0.52848$ or about $52.85\%$