91影视

Probability is the likelihood that an event will occur and is calculated by dividing the number of favourable outcomes by the total number of possible outcomes. The simplest example is a coin flip. When you flip a coin there are only two possible outcomes, the result is either heads or tails.

(a)

Given: \({\rm{X}}\) has a gamma distribution with

\(\mu = 37.5\)

\(\sigma = 21.6\)

A gamma distribution's mean and variance are calculated as follows:

\(\mu = E(X) = \alpha \beta \)

\({\sigma ^2} = V(X) = \alpha {\beta ^2}\)

The mean is \(37.5\)dollars, and the standard deviation is \(21.6\)dollars:

\(\,37.5 = \alpha \beta \)

\({21.6^2} = \alpha {\beta ^2}\,\,\)

DETERMINE \(\beta \)

Use in the second equation the fact that \(\alpha \beta = 37.5\) (first equation):

\(466.56 = {21.6^2} = \alpha \beta = \alpha \beta \times \beta = 37.5 \times \beta \)

Divide each side by \(37.5\):

\(\frac{{466.56}}{{37.5}} = \beta \)

Replace the left and right sides and assess:

\(\beta = \frac{{466.56}}{{37.5}} = 12.4416\)

DETERMINE \({\bf{\alpha }}\)

Equation found for the mean:

\(37.5 = \alpha \beta \)

Replace \(\beta \)by\(12.4416\):

\(37.5 = \alpha 12.4416\)

Divide each side by\(12.4416\):

\(\frac{{37.5}}{{12.4416}} = \alpha \)

Interchange the left and right side and evaluate:

\(\alpha = \frac{{37.5}}{{12.4416}} \approx 3.0141\)

Therefore, the value of \(\alpha \) and \(\beta \)is \(\alpha = 3.0141\), \(\beta = 12.4416\).

(b)

The gamma distribution of \(X\) is standard.

\(\alpha = 3.0141\)

\(\beta = 12.4416\)

\({\rm{P(X > 50)}}\)

Complement rule:

\(P({\mathop{\rm not}\nolimits} {\rm{A}}) = 1 - P(A)\)

Gamma distribution of a property:

\(P(X \le x) = F(x;\alpha ,\beta ) = F\left( {\frac{x}{\beta },\alpha } \right)\)

The incomplete gamma function is represented by \(F.\)

For \(x = 50\)and the complement rule, use this property:

\(P(X > 50) = 1 - P(X \le 50) = 1 - F\left( {\frac{{50}}{{12.4416}},3.0141} \right) \approx 1 - F(4.0188,3.0141) \approx 1 - F(4;3)\)

The incomplete gamma functions with \(x = 4\)and \(\alpha = 3\) is \(F(4;3).\)

The value can be found in the appendix's incomplete gamma function table (in the column with \(\alpha = 3\) and in the row with\(x = 4\)).

\(F(4;3) = 0.762\)

The probability then becomes:

\(P(X > 50) = 1 - F(4;3) = 1 - 0.762 = 0.238 = 23.8\% \)

Therefore, the probability of that data transfer time exceeds \(50{\rm{ }}ms\) is \(P(X > 50) = 0.238 = 23.8\% \).

(c)

Given: The gamma distribution of \(X\) is standard.

\(\alpha = 3.0141\)

\(\beta = 12.4416\)

\({\rm{P(50 < X < 75)}}\)

Property gamma distribution:

\(P(X \le x) = F(x;\alpha ,\beta ) = F\left( {\frac{x}{\beta },\alpha } \right)\)

The incomplete gamma function is represented by \(F.\)

For \(x = 50\) and \(x = 75,\)use this property:

\(P(X \le 50) = F\left( {\frac{{50}}{{12.4416}},3.0141} \right) \approx F(4.0188,3.0141) \approx F(4;3)\)

\(P(X \le 75) = F\left( {\frac{{75}}{{12.4416}},3.0141} \right) \approx F(6.0282,3.0141) \approx F(6;3)\)

The incomplete gamma function with \(x = 4\)and \(\alpha = 3\)is \(F(4;3)\).

In the appendix's incomplete gamma function table (table A.4), the value can be found in the row with \(x = 4\) and the column with \(\alpha = 3\):

\(F(4;3) = 0.762\)

The incomplete gamma functions with \(x = 6\)and \(\alpha = 3\) is \(F(6;3)\)

The value may be found in the appendix's incomplete gamma function table (in the column with \(\alpha = 3\)and in the row with \(x = 6\)):

\(F(6;3) = 0.938\)

The difference in probabilities to the left of two boundaries is the probability between them:

\(P(50 < X < 75) = P(X < 75) - P(X \le 50) = P(X \le 75) - P(X \le 50)\)

\({\rm{ = F(6 ; 3) - F(4 ; 3) = 0}}{\rm{.938 - 0}}{\rm{.762 = 0}}{\rm{.176 = 17}}{\rm{.6 \backslash \% }}\)

Therefore, the probability of that data transfer time is between \(50\) and \(75{\rm{ }}ms\) is \(P(50 < X < 75) = 0.176 = 17.6\% \).