Respuesta :
Answer:
Explained below.
Step-by-step explanation:
The random variable X is defined as the number of missing pulses and follows a Poisson distribution with parameter (μ = 0.50).
The probability mass function of X is as follows:
[tex]P(X=x)=\frac{e^{-\mu}\ \mu^{x}}{x!};\ x=0,1,2,3...[/tex]
(a)
Compute the probability that a disk has exactly one missing pulse as follows:
[tex]P(X=1)=\frac{e^{-0.50}\ 0.50^{1}}{1!}=0.3033[/tex]
Thus, the probability that a disk has exactly one missing pulse is 0.3033.
(b)
Compute the probability that a disk has at least two missing pulses as follows:
[tex]P(X\geq 2)=1-P(X<2)\\[/tex]
[tex]=1-[P(X=0)+P(X=1)]\\=1-[\frac{e^{-0.50}\ 0.50^{0}}{0!}+\frac{e^{-0.50}\ 0.50^{1}}{1!}]\\=1-0.6065-0.3033\\=0.0902[/tex]
Thus, the probability that a disk has at least two missing pulses is 0.0902.
(c)
It is provided that the two disks selected are independent of each other.
The probability that a disk has no missing pulses is:
[tex]P(X=0)=\frac{e^{-0.50}\ 0.50^{0}}{0!}=0.6065[/tex]
Compute the probability that neither of the two disks contains a missing pulse as follows:
[tex]P(X_{1}=0,\ X_{2}=0)=P(X_{1}=0)\times P(X_{2}=0)[/tex]
[tex]=0.6065\times 0.6065\\=0.367842\\\approx 0.3678[/tex]
Thus, the probability that neither of the two disks contains a missing pulse is 0.3678.
