# calculating probability of a return below a specific value [closed]

assume a probability distribution with a mean of %10 and standard deviation of %1.5. In wanting to solve the probability being lower than %5, the normal distribution is written down and integrated as follows:

$$\int_{-∞}^{0.05} e^{(-\frac{1}{2} ((x - 0.1)/0.015)^2)}/(\sqrt{2 π} 0.015) dx = 0.00042906$$

which gives %0.04

if you calculate it through the z score(assuming a sample size of 10) it would be:

Z=(0.05-0.10)/(0.015/sqrt(10))=-10.5409

looking at a z score table would then give a probability of 2.798*10^-26

why do the two approaches give different answeres?

• Where does the sqrt(10) come from? Clearly, $$z\equiv \frac{x-\mu}{\sigma}=-10/3$$, and $N(z)=0.0042906$ Aug 23, 2022 at 9:20
• "standard deviation of %1.5" is unusually small. Are you sure of this value? I think 15% (or more) would be a more usual value for stock returns. No surprise then that the probability you found is quite low. Any outcome outside of $\mu \pm 2 \sigma$ is rare. Aug 23, 2022 at 9:57