# Compute the probability under the normal distribution

Given:

1) Invest $10,000 in Stock "ABC".

2) The daily returns are normally distributed with a mean of 0% and a standard deviation of 1%.

Questions:

1) What is the probability that tomorrow's return on investment in ABC is exactly 0.9%?

2) Conditional on making a positive return tomorrow, what is the probability you make a return of at least 1%?

3) What is the probability that tomorrow's return on your investment in ABC is somewhere between -1% and +.5%?

4) Calculate the "value at risk" for investment of $20,000 at the 95% confidence level.

5) You believe that 1% of mutual fund managers can "beat the market" and deliver a positive mean daily return of .5%, also with a standard deviation of 1%. The remaining 99% of funds cannot beat the market; they all have identical daily mean return of 0% and a standard deviation of 1%. A broker calls you and claims his fund is one that can beat the market. You follow the fund for the day and see that today it has a positive (greater than 0%) return. Use Bayes Theorem to compute the probability that this fund can actually beat the market.

https://brainmass.com/math/probability/compute-probability-under-normal-distribution-597870

#### Solution Preview

1) Since the dairy returns are normally distributed, it is a continuous distribution. So the probability that tomorrow's return on investment in ABC is exactly 0.9% is zero.

2) Since ...

#### Solution Summary

The solution gives detailed steps on computing the probability under the normal distribution and the conditional probability applying Bayes theorem.