Respuesta :
Answer:
a) 0.0821
b) 0.0111
c) 0.0041
Step-by-step explanation:
Exponential distribution:
The exponential probability distribution, with mean m, is described by the following equation:
[tex]f(x) = \lambda e^{-\lambda x}[/tex]
In which [tex]\lambda[/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^{-\lambda x}[/tex]
Either it lasts more 5 years or less, or it survives more than 5 years. The sum of the probabilities of these events is decimal 1. So
[tex]P(X \leq 5) + P(X > 5) = 1[/tex]
In all 3 cases, we want P(X > 5). So
[tex]P(X > 5) = 1 - P(X \leq 5)[/tex]
In which
[tex]P(X \leq 5) = 1 - e^{-5\lambda}[/tex]
(a) lambda=.5
[tex]P(X \leq 5) = 1 - e^{-5*0.5} = 0.9179[/tex]
[tex]P(X > 5) = 1 - P(X \leq 5) = 1 - 0.9179 = 0.0821[/tex]
(b) lambda=0.9
[tex]P(X \leq 5) = 1 - e^{-5*0.9} = 0.9889[/tex]
[tex]P(X > 5) = 1 - P(X \leq 5) = 1 - 0.9889 = 0.0111[/tex]
(c) lambda=1.1
[tex]P(X \leq 5) = 1 - e^{-5*1.1} = 0.9959[/tex]
[tex]P(X > 5) = 1 - P(X \leq 5) = 1 - 0.9959 = 0.0041[/tex]
