Question

Data collected over time on the utilization of a computer core (as a proportion of the total capacity) were found to possess a relative frequency distribution that could be approximated by a beta density function with α = 2 and β = 4. Find the probability that the proportion of the core being used at any particular time will be less than 0.10.

Answer 1

The probability that the proportion of the core being used at any particular time will be less than 0.10 is 0.08146

Explanation:

`\mu =(\alpha )/(\alpha +\beta ) = (1)/(1 + (\beta)/(\alpha) )` 1/3 0.33 = 33.33 %

The Probability of that the proportion of the core being used at any particular time will be less than 0.10 is given by

PDF =
`\frac{x^(\alpha -1) (1-x)^(\beta -1) }{\int\limits^1_0 {u^(\alpha -1) (1-u)^(\beta -1)} \, du }`

where x = 0.1

α = 2 and β = 4

PDF =
`\frac{0.0729 }{\int\limits^1_0 {u^(\alpha -1) (1-u)^(\beta -1)} \, du }` = 1.458

CDF =
`\frac{\int\limits^(0.1)_0 {t^(\alpha -1) (1-t)^(\beta -1)} \, du }{\int\limits^1_0 {u^(\alpha -1) (1-u)^(\beta -1)} \, du }` = 0.08146

The probability that the proportion of the core being used at any particular time will be less than 0.10 = 0.08146.