Respuesta :
Answer:
1) [tex]P(x=0) =[/tex] 0.0003355
2) [tex]P(U|W) =[/tex] 0.000838
Step-by-step explanation:
The number of emails received on Sunday can be modeled by a Poisson distribution with an average of 1/30 emails per minute.
Also; taking the probability that no email in an interval of the length of 4 hours on Sunday; we have:
Time (T) = 4 hours
= 4 × 60 minutes
= 240 minutes
Poisson Distribution can be expressed as:
[tex]\lambda = T * \frac{1}{30}[/tex]
[tex]\lambda=[/tex] [tex]240*\frac{1}{30}[/tex]
[tex]\lambda=[/tex] 8
The Poisson Probability can now be calculated by using the formula;
[tex]P(x) = \frac{e^ {- \lambda} \lambda^x}{x!}[/tex]
[tex]P(x=0) = \frac{e^ {8} \lambda^0}{0!}[/tex]
[tex]P(x=0) =[/tex] [tex]e^{-8}[/tex]
= 0.0003355
2)
A random day is chosen (all days of the week are equally likely to be selected), and a random interval of length one hour is selected on the chosen day. It is observed that I did not receive any emails in that interval.
What is the probability that the chosen day is a weekday?
Let use U to represent the event that weekday(Monday - Friday) is chosen.
Hence, P(U) = [tex]\frac{5}{7}[/tex]
Let V represent the event that weekend( Saturday or Sunday ) is chosen.
P(V) = [tex]\frac{2}{7}[/tex]
Let W represent the event that no emails were received.
The probability that no emails were received during the weekday can then be illustrated as:
P ( W║U)
The Poisson Distribution for weekday (Monday - Friday ) is given by:
[tex]\lambda = \frac{1}{6} *1 hour[/tex]
[tex]\lambda = \frac{60}{6}minutes[/tex]
[tex]\lambda = 10 minutes[/tex]
Probability P ( W║U) for Poisson Probability can now be determined as:
[tex]P(x) = \frac{e^ {- \lambda} \lambda^x}{x!}[/tex]
[tex]P(x=0) = \frac{e^ {-10} \lambda^0}{0!}[/tex]
[tex]P(x=0) =[/tex] [tex]e^{-10}[/tex]
Probability that no emails we received during the weekend (Saturday or Sunday) can be illustrated as: P ( W║V).
The Poisson distribution for weekend (Saturday or Sunday) is given by:
[tex]\lambda = \frac{1}{30} *1 hour[/tex]
[tex]\lambda = \frac{60}{30}minutes[/tex]
[tex]\lambda = 2 minutes[/tex]
P ( W║V) for Poisson Probability can be be calculated as:
[tex]P(x) = \frac{e^ {- \lambda} \lambda^x}{x!}[/tex]
[tex]P(x=0) = \frac{e^ {-2} \lambda^0}{0!}[/tex]
[tex]P(x=0) =[/tex] [tex]e^{-2}[/tex]
FInally, the probability that the chosen day is a weekday given that no emails were received in the interval of the length one hour can now be determined as:
[tex]P ( U|W) =\frac{P(W|U)P(U)}{P(W)}[/tex]
[tex]P ( U|W) =\frac{P(W|U)P(U)}{P(W|U)P(U)+P(W|V)P(V)}[/tex]
[tex]P(U|W) =[/tex] [tex]\frac{e^{-10}*\frac{5}{7} }{(e^{-10}*\frac{5}{7})+(e^{-2}*\frac{2}{7}) }[/tex]
[tex]P(U|W) =[/tex] [tex]\frac{5e^{-10}}{5e^{-10}+2e^{-2}}[/tex]
[tex]P(U|W) =[/tex] [tex]\frac{5e^{-10}}{e^{-10}(2+e^{8})}[/tex]
[tex]P(U|W) =[/tex] [tex]\frac{5}{(5+2e^8)}[/tex]
[tex]P(U|W) =[/tex] 0.000838
- The probability that you get no emails in an interval of length 4 hours on a Sunday is [tex]3.36 \times 10^{-4}[/tex].
- The probability that the chosen day is a weekday is [tex]8.38 \times 10^{-4}[/tex].
Given the following data:
- Time = 4 hours to minutes = [tex]4 \times 60[/tex] = 240 minutes.
- Poisson distribution (Monday through Friday) = [tex]\frac{1}{6}[/tex] emails per minute.
- Poisson distribution (Saturday and Sunday) = [tex]\frac{1}{30}[/tex] emails per minute.
How to calculate Poisson distribution.
Mathematically, Poisson distribution for the emails received on Saturday and Sunday is given by this formula:
[tex]\lambda = T \times \frac{1}{30} \\\\\lambda = 240 \times \frac{1}{30} \\\\\lambda = 8[/tex]
For the Poisson probability:
Mathematically, Poisson probability is given by this formula:
[tex]P(x) = \frac{e^{-\lambda}\lambda^x}{x!} \\\\P(x=0) = \frac{e^{-8}\times 8^0}{0!} \\\\P(x=0) =e^{-8}\\\\P(x=0) =3.36 \times 10^{-4}[/tex]
b. To calculate the probability that the chosen day is a weekday:
Time = 1 hour = 60 minutes.
- Let M denote that (Monday through Friday) is chosen.
[tex]P(M)=\frac{5}{7}[/tex]
- Let S denote that (Saturday and Sunday) is chosen.
[tex]P(S)=\frac{2}{7}[/tex]
- Let N denote when no email is received.
For the Poisson distribution P (N||M):
[tex]\lambda = T \times \frac{1}{6} \\\\\lambda = 60 \times \frac{60}{6} \\\\\lambda = 10[/tex]
For the Poisson probability P (N||M):
[tex]P(x) = \frac{e^{-\lambda}\lambda^x}{x!} \\\\P(x=0) = \frac{e^{-10}\times 10^0}{0!} \\\\P(x=0) =e^{-10}[/tex]
For the Poisson distribution P (N||S):
[tex]\lambda = T \times \frac{1}{30} \\\\\lambda = 60 \times \frac{60}{30} \\\\\lambda = 10[/tex]
For the Poisson probability P (N||S):
[tex]P(x) = \frac{e^{-\lambda}\lambda^x}{x!} \\\\P(x=0) = \frac{e^{-2}\times 2^0}{0!} \\\\P(x=0) =e^{-2}[/tex]
Now, we can calculate the probability that the chosen day is a weekday:
[tex]P(M||N) = \frac{P(M||N)P(M)}{P(N)} \\\\P(M||N) = \frac{P(M||N)P(M)}{P(M||N)P(M)+P(S||N)P(S)}[/tex]
Substituting the parameters into the formula, we have;
[tex]P(M||N) = \frac{e^{-10} \times \frac{5}{7} }{e^{-10} \times \frac{5}{7}\; +\;e^{-2} \times \frac{2}{7} }\\\\P(M||N) = \frac{5e^{-10} }{5e^{-10} \; +\;2e^{-2} }\\\\P(M||N) = \frac{5e^{-10} }{5e^{-10} \; +\;2e^{-2} }\\\\P(M||N) = 8.38 \times 10^{-4}[/tex]
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