Respuesta :
Answer:
(a) The value of P (X > 10) is 0.3679.
(b) The value of P (X > 20) is 0.1353.
(c) The value of P (X < 30) is 0.9502.
(d) The value of x is 30.
Step-by-step explanation:
The probability density function of an exponential distribution is:
[tex]f(x)=\lambda e^{-\lambda x};\ x>0, \lambda>0[/tex]
The value of E (X) is 10.
The parameter λ is:
[tex]\lambda=\frac{1}{E(X)}=\frac{1}{10}=0.10[/tex]
(a)
Compute the value of P (X > 10) as follows:
[tex]P(X>10)=\int\limits^{\infty}_{10} {0.10 e^{-0.10 x}} \, dx \\=0.10\int\limits^{\infty}_{10} { e^{-0.10 x}} \, dx\\=0.10|\frac{e^{-0.10 x}}{-0.10} |^{\infty}_{10}\\=|e^{-0.10 x} |^{\infty}_{10}\\=e^{-0.10\times10}\\=0.3679[/tex]
Thus, the value of P (X > 10) is 0.3679.
(b)
Compute the value of P (X > 20) as follows:
[tex]P(X>20)=\int\limits^{\infty}_{20} {0.10 e^{-0.10 x}} \, dx \\=0.10\int\limits^{\infty}_{20} { e^{-0.10 x}} \, dx\\=0.10|\frac{e^{-0.10 x}}{-0.10} |^{\infty}_{20}\\=|e^{-0.10 x} |^{\infty}_{20}\\=e^{-0.10\times20}\\=0.1353[/tex]
Thus, the value of P (X > 20) is 0.1353.
(c)
Compute the value of P (X < 30) as follows:
[tex]P(X<30)=\int\limits^{30}_{0} {0.10 e^{-0.10 x}} \, dx \\=0.10\int\limits^{30}_{0} { e^{-0.10 x}} \, dx\\=0.10|\frac{e^{-0.10 x}}{-0.10} |^{30}_{0}\\=|e^{-0.10 x} |^{30}_{0}\\=1-e^{-0.10\times30}\\=1-0.0498\\=0.9502[/tex]
Thus, the value of P (X < 30) is 0.9502.
(d)
It is given that, P (X < x) = 0.95.
Compute the value of x as follows:
[tex]P(X<x)=0.95\\\int\limits^{x}_{0} {0.10 e^{-0.10 x}} \, dx=0.95\\0.10\int\limits^{x}_{0} { e^{-0.10 x}} \, dx=0.95\\0.10|\frac{e^{-0.10 x}}{-0.10} |^{x}_{0}=0.95\\|e^{-0.10 x} |^{x}_{0}=0.95\\1-e^{-0.10\times x}=0.95\\e^{-0.10\times x}=0.05[/tex]
Take natural log on both sides.
[tex]ln(e^{-0.10x})=ln(0.05)\\-0.10x=-2.996\\x=\frac{2.996}{0.10}\\ =29.96\\\approx30[/tex]
Thus, the value of x is 30.
