Q. 69 - Calculus

Question

Many email filters can be trained how to recognize spam by having a user identify spam messages from a lineup. Suppose you have looked at 12 messages with the word “opportunity” in the subject line, and identified 8 of them to the filter as spam. Now you get a new message with that word in the subject line, and your filter must compute the probability p that the message is spam. One way to do that is to identify p as the expected value of the Beta distribution 

$f\left(x\right)=6435x^8{(1-x)}^4$

The expected value of this distribution is ${\int }_0^{1}xf\left(x\right)dx$

Evaluate the probability that the next message you receive with the word “opportunity” in the subject line will be spam 

Step-by-Step Solution

Answer

The probability that the next message you received with "opportunity"is $\frac{9}{14}s$

Step 1. Given information

The given beta distribution $f\left(x\right)=6435x^8{(1-x)}^4$

Step 2. Evaluate the probability of given function

The expected value of the distribution is,

$$\begin{gathered} {\int }_0^{1}xf\left(x\right)dx={\int }_0^{1}x.6435x^8{(1-x)}^{4}dx \\ =6435{\int }_0^1{x}^9{(1-x)}^{4}dx \\ =6435{\int }_0^1{x}^9(1-4x+6x^2-4x^3+x^4)dx \\ =6435{\int }_0^1(x^9-4x^{10}+6x^{11}-4x^{12}+x^{13})dx \\ ={6435\left(\frac{x^{10}}{10}-4\frac{x^{11}}{11}+6\frac{x^{12}}{12}-4\frac{x^{13}}{13}+\frac{x^{14}}{14}\right)}_0^1 \\ =6435\left(\frac{1}{10}-\frac{4}{11}+\frac{6}{12}-\frac{4}{13}+\frac{1}{14}\right) \\ =\frac{9}{14} \end{gathered}$$

Thus,the probability that the next message you received with "opportunity"is $\frac{9}{14}s$