91影视

Probability denotes the possibility of something happening. It's a field of mathematics that studies the probability of a random event occurring.

a)

Let

\(X = \)number of incoming calls that involve fax messages.

Since \(25\% \) of the incoming calls involve fax messages, \(p = 0.25\) and there are \(n = 25\) incoming calls. Therefore, \(X\sim Bin(25,0,25\)) (Binomial Distribution).

(a):

Cumulative Density Function cdf of binomial random variable X with parameters n and p is

\(\begin{array}{c}B(x;n,p) = P(X \le x)\\ = \sum\limits_{y = 0}^x b (y;n,p),\;\;\;\\x = 0,1, \ldots ,n.\end{array}\)

Therefore, the following is true

\(\begin{array}{c}B(6;25,0.25) = P(X \le 6)\\\mathop = \limits^{(1)} 0.5611.\end{array}\)

Therefore, the value can be found in Appendix Table A.l. (column " \(0.25\)鈥� and row " 6".

b)

Theorem:

\(b(x;n,p) = \left\{ {\begin{array}{*{20}{l}}{\left( {\begin{array}{*{20}{l}}n\\x\end{array}} \right){p^x}{{(1 - p)}^{n - x}}}&{,x = 0,1,2, \ldots ,n}\\0&{\rm{ }}\end{array}} \right.\)

The following is correct

\(\begin{array}{c}P(X = 6)\\ = b(6;25,0.25)\\ = 256\end{array}\)

Therefore, \({0.25^6}{(1 - 0.25)^{25 - 6}} = 0.1828.\)

c)

The following holds

\(\begin{array}{c}P(X \ge 6)\mathop = \limits^{(1)} 1\\ - P(X < 6)\mathop = \limits^{(2)} 1\\ - P(X \le 5) = 1\\ - B(5;25,0.25)\mathop = \limits^{(3)} 1\\ - 0.3783 = 0.6217.\end{array}\)

Complement of event\(\{ X \ge 6\} \)is event\(\{ X < 6\} \);

X takes only non-negative values;

As a result, the value is in Appendix Table A.l. (column " \({0.25^\circ }\)and row "5").

d)

The following is true

\(\begin{array}{c}P(X > 6)\mathop = \limits^{(1)} 1 - P\\(X \le 6) = 1 - 0.5611\\ = 0.4389\end{array}\)

Therefore, complement of event \(\{ X > 6\} \)is event \(\{ X \le 6\} \).