# Calculate Quarter profit probability if you know daily average and standard deviation

A company's daily profit is on average is €$100,000$ And the standard deviation is €$10,000$. What is the probability of the company in Quarter to show a profit of more than € $9.2$ million?

The Quarter average will be $100,000 * 90=9,000,000$ What about the standard deviation it will be $10,000*8=90$ or something else.

• Assuming the quarter is 92 days, the standard deviation does not matter for the answer. – kludg Jun 14 '17 at 6:35
• Propability $\to$ probability – Jean Marie Jun 14 '17 at 6:36

Assuming that the daily profits are independent and identically distributed you can apply the central limit theorem.

Let $X_i$ be the random variable for the daily profit. Then the random variable for the sum of the daily profit is $Y_{90}=\sum\limits_{i=1}^{90} X_i$. The expeted value of $Y_{90}$ is $90\cdot E(X_i)=90\cdot 100,000=9,000,000$

The variance of $Y_{90}$ is $\sum_{i=1}^{90} Var(X_i)=90\cdot Var(X_i)=90\cdot 10,000^2$.

Thus the standard deviation is $\sqrt{Var(Y_{90})}=\sqrt{90}\cdot 10,000$. Therefore

$$P(Y_{90}>9,2 \ \texttt{Mio})=1-P(Y_{90}<9,2 \ \texttt{Mio})$$

$$\approx 1- \Phi\left( \frac{9,200,000-9,000,000}{\sqrt{90}\cdot 10,000} \right)=1-\Phi(2.108)=1-0.9825=0.0175=1.75\%$$

$\Phi(z)$ is the cdf of the standard normal distribution.