Respuesta :
Answer:
The quartile 3 is 0.3
Step-by-step explanation:
Exponential distribution:
The exponential probability distribution, with mean m, is described by the following equation:
[tex]f(x) = \mu e^{-\mu x}[/tex]
In which [tex]\mu = \frac{1}{m}[/tex] is the decay parameter.
The probability that x is lower or equal to a is given by:
[tex]P(X \leq x) = \int\limits^a_0 {f(x)} \, dx[/tex]
Which has the following solution:
[tex]P(X \leq x) = 1 - e^{-\mu x}[/tex]
The probability of finding a value higher than x is:
[tex]P(X > x) = 1 - P(X \leq x) = 1 - (1 - e^{-\mu x}) = e^{-\mu x}[/tex]
X~Exp (0.2)
This means that [tex]m= 0.2, \mu = \frac{1}{0.2} = 5[/tex]
Find the quartile 3.
The 3rd quartile is the 75th percentile, for which [tex]P(X \leq x) = 0.75[/tex], or [tex]P(X > x) = 1 - 0.75 = 0.25[/tex]
Since
[tex]P(X > x) = e^{-\mu x}[/tex]
[tex]e^{-5x} = 0.25[/tex]
[tex]\ln{e^{-5x}} = \ln{0.25}[/tex]
[tex]-5x = \ln{0.25}[/tex]
[tex]x = -\frac{\ln{0.25}}{5}[/tex]
[tex]x = 0.277[/tex]
Rounding to the nearest tenth, the quartile 3 is 0.3.
