The question: Example of Solution:
Posted: Wed May 11, 2022 6:44 am
The question:
Example of Solution:
The number of students enrolled in calculus classes at FIT is a Poisson r.v. with parameter 1 = 144. Use the CLT approximation to find the probability that the new enrollment is going to be 168 or more students. Also show what the exact solution would have been without the CLT approximation. Explain your steps. Hint. Use Table 1 (Gaussian PDF). (35 pts)
2 The number of students enrolled in calculus classes at FIT is a Poisson r.v. with parameter 1 = 100. Use the CLT approximation to find the probability that the new enrollment is going to be 120 or more students. Solution. The exact solution, without the approximation is P{X > 120} = 1 -e-100 Σ 119 100% k=0 which numerically is too cumbersome. Recalling that a Poisson distribution is "infinitely divisible" we can represent X as any sum of iid Poisson r.v.'s each with parameter, say /n. For instance we can use n= 100. Hence we can apply the CLT directly to X pretending that it is a sum, like in Remark 2.2 or Example 2.3. Recalling that the mean of X is 100 and so is the variance, we have using Gaussian Table 1 p{ X-100 100 $ 120-100 10 } }~1 – Þ(2) = 0.0228.
- The Question Example Of Solution 1 (29.72 KiB) Viewed 29 times
- The Question Example Of Solution 2 (49.73 KiB) Viewed 29 times
2 The number of students enrolled in calculus classes at FIT is a Poisson r.v. with parameter 1 = 100. Use the CLT approximation to find the probability that the new enrollment is going to be 120 or more students. Solution. The exact solution, without the approximation is P{X > 120} = 1 -e-100 Σ 119 100% k=0 which numerically is too cumbersome. Recalling that a Poisson distribution is "infinitely divisible" we can represent X as any sum of iid Poisson r.v.'s each with parameter, say /n. For instance we can use n= 100. Hence we can apply the CLT directly to X pretending that it is a sum, like in Remark 2.2 or Example 2.3. Recalling that the mean of X is 100 and so is the variance, we have using Gaussian Table 1 p{ X-100 100 $ 120-100 10 } }~1 – Þ(2) = 0.0228.