Answer:
a) P(x=3)=0.089
b) P(x≥3)=0.938
c) 1.5 arrivals
Step-by-step explanation:
Let t be the time (in hours), then random variable X is the number of people arriving for treatment at an emergency room.
The variable X is modeled by a Poisson process with a rate parameter of λ=6.
The probability of exactly k arrivals in a particular hour can be written as:
![P(x=k)=\lambda^{k} \cdot e^{-\lambda}/k!\\\\P(x=k)=6^k\cdot e^{-6}/k!](https://tex.z-dn.net/?f=P%28x%3Dk%29%3D%5Clambda%5E%7Bk%7D%20%5Ccdot%20e%5E%7B-%5Clambda%7D%2Fk%21%5C%5C%5C%5CP%28x%3Dk%29%3D6%5Ek%5Ccdot%20e%5E%7B-6%7D%2Fk%21)
a) The probability that exactly 3 arrivals occur during a particular hour is:
![P(x=3)=6^{3} \cdot e^{-6}/3!=216*0.0025/6=0.089\\\\](https://tex.z-dn.net/?f=P%28x%3D3%29%3D6%5E%7B3%7D%20%5Ccdot%20e%5E%7B-6%7D%2F3%21%3D216%2A0.0025%2F6%3D0.089%5C%5C%5C%5C)
b) The probability that <em>at least</em> 3 people arrive during a particular hour is:
![P(x\geq3)=1-[P(x=0)+P(x=1)+P(x=2)]\\\\\\P(0)=6^{0} \cdot e^{-6}/0!=1*0.0025/1=0.002\\\\P(1)=6^{1} \cdot e^{-6}/1!=6*0.0025/1=0.015\\\\P(2)=6^{2} \cdot e^{-6}/2!=36*0.0025/2=0.045\\\\\\P(x\geq3)=1-[0.002+0.015+0.045]=1-0.062=0.938](https://tex.z-dn.net/?f=P%28x%5Cgeq3%29%3D1-%5BP%28x%3D0%29%2BP%28x%3D1%29%2BP%28x%3D2%29%5D%5C%5C%5C%5C%5C%5CP%280%29%3D6%5E%7B0%7D%20%5Ccdot%20e%5E%7B-6%7D%2F0%21%3D1%2A0.0025%2F1%3D0.002%5C%5C%5C%5CP%281%29%3D6%5E%7B1%7D%20%5Ccdot%20e%5E%7B-6%7D%2F1%21%3D6%2A0.0025%2F1%3D0.015%5C%5C%5C%5CP%282%29%3D6%5E%7B2%7D%20%5Ccdot%20e%5E%7B-6%7D%2F2%21%3D36%2A0.0025%2F2%3D0.045%5C%5C%5C%5C%5C%5CP%28x%5Cgeq3%29%3D1-%5B0.002%2B0.015%2B0.045%5D%3D1-0.062%3D0.938)
c) In this case, t=0.25, so we recalculate the parameter as:
![\lambda =r\cdot t=6\;h^{-1}\cdot 0.25 h=1.5](https://tex.z-dn.net/?f=%5Clambda%20%3Dr%5Ccdot%20t%3D6%5C%3Bh%5E%7B-1%7D%5Ccdot%200.25%20h%3D1.5)
The expected value for a Poisson distribution is equal to its parameter λ, so in this case we expect 1.5 arrivals in a period of 15 minutes.
![E(x)=\lambda=1.5](https://tex.z-dn.net/?f=E%28x%29%3D%5Clambda%3D1.5)
Answer:
x-1
Step-by-step explanation:
if "a number" is x, then one less than that would be subtracting one.
= x-1
Hope this helps!
Answer:
no
Step-by-step explanation: