Respuesta :
Answer:
(a) The probability that the lifetime is at most 2000 h is 0.8647.
(b) The probability that the lifetime is at most 2000 h is 0.8647.
(c) The probability that the lifetime is between 500 h and 2000 h is 0.4712.
(d) The variance of the lifetime of a particular component is 10⁻⁶.
Step-by-step explanation:
Let X = lifetime of a particular component
The random variable [tex]X\sim Exp(\lambda = \frac{1}{1000} )[/tex]
The probability distribution function of an exponential distribution is:
[tex]f(x)=\left \{ {{\lambda e^{-\lambda x}; x>0\atop {0};\ otherwise} \right.[/tex]
(a)
Compute the probability that the lifetime is at most 2000 h as follows:
[tex]P(X\leq 2000)=\int\limits^{2000}_{0} {\lambda e^{-\lambda x}} \, dx \\=\lambda[\frac{e^{-\lambda x}}{-\lambda} ]^{2000}_{0} \\=[-e^{\frac{-2000}{1000}}+e^{\frac{-0}{1000} } }]\\=1-0.1353\\=0.8647[/tex]
Thus, the probability that the lifetime is at most 2000 h is 0.8647.
(b)
Compute the probability that the lifetime is more than 1000 h as follows:
[tex]P(X>1000)=\int\limits^{\infty}_{1000} {\lambda e^{-\lambda x}} \, dx \\=\lambda[\frac{e^{-\lambda x}}{-\lambda} ]^{\infty}_{1000} \\=[-e^{\frac{-\infty}{1000}}+e^{\frac{-1000}{1000} } }]\\=0+0.3679\\=0.3679[/tex]
Thus, the probability that the lifetime is more than 1000 h is 0.3679.
(c)
Compute the probability that the lifetime is between 500 h and 2000 h as follows:
[tex]P(500<X<2000)=\int\limits^{2000}_{500} {\lambda e^{-\lambda x}} \, dx \\=\lambda[\frac{e^{-\lambda x}}{-\lambda} ]^{2000}_{500} \\=[-e^{\frac{-2000}{1000}}+e^{\frac{-500}{1000} } }]\\=-0.1353+0.6065\\=0.4712[/tex]
Thus, the probability that the lifetime is between 500 h and 2000 h is 0.4712.
(d)
The variance of an exponential distribution is, [tex]Var(X)=\frac{1}{\lambda^{2}}[/tex]
The variance of the lifetime of a particular component is:
[tex]Var(X)=\frac{1}{\lambda^{2}}=(\frac{1}{\lambda})^{2}=(\frac{1}{1000} )^{2}=10^{-6}[/tex]
Thus, the variance of the lifetime of a particular component is 10⁻⁶.
