1. Consider three binomial distributions with n=5 and p=0.8, 0.5, and 0.2. Plot the pmf of the three distributions. Draw random samples of size 100 from each of the three distributions using the rbinom function and plot the relative frequency distributions. 2. A hospital administrator, who has been studying daily emergency admissions over a period of several years, has concluded that they are distributed according to the Poisson law. Hospital records reveal that emergency admissions have average three per day during this period. If the administrator is correct in assuming a Poisson distribution nd the probability that (a) Exactly two emergency admissions will occur on a given day. (b) No emergency admissions will occur on a particular day. 3. Simulate 500 samples from a Normal population having mean 20 and sd 5. 4. Simulate 200 samples from standard normal population 5. It has been suggested IQ scores follow a normal distribution with mean 99 and standard deviation 17. Find the probability that any person chosen at random will have (a) An IQ greater than 105 (b) An IQ less than 65 using R

Exercise: Emergency hospital admissions in USA with herat disease are found to followPoisson distribution. An investigation showed that there are four admissions per day(mean). Find the probability that exactly three emergency admissions will occur in a givenday?

3. At Kabwe General Hospital, an average of 6 accident victims arrive per night in the intensive care unit. However, arrangements have been made such that the intensive care unit is able to handle 8 accident victims per night. If this number is exceeded the patients will not get optimal treatment as the unit’s capacity will have been exceeded. For the purpose of this exercise, let’s assume that X = number of accident victims who arrive at the intensive care unit per night is Poisson distributed with λ = 6. a) Is the assumption that X = number of accident victims who arrive at the intensive care unit per night is Poisson distributed a valid one? Give augments in favour and against this assumption. b) Given a night, what is the probability that the unit’s capacity will be exceeded? c) Given 10 nights, what is the probability that the capacity of the unit will be exceeded? d) What is the probability of at most 8 accident victims arriving at the unit in a given night?

2. The number of accidents (A) at an intersection is counted for a 12-hour period. The number of accidents follows a Poisson distribution with a parameter value of 3.5. (Round to 3 decimal places and use a point-not a comma).Calculate the following:a. 𝑃(𝐴 = 1) Blank 1b. 𝑃(𝐴 < 5) Blank 2c. 𝑃(𝐴 > 0) Blank 3d. 𝑃(0 < 𝐴 ≤ 3) Blank 4e. The probability that there will be 4 accidents in a 24-hour period Blank 5f. The expected number of accidents per week. Blank 6g. Let 𝐵=number of hours in a 12-hour cycle that there are no accidents. Calculate the probability that for 8 hours in the 12-hour cycle there will be no accidents. Blank 7

The number of accidents (𝐴) at an intersection is counted for a 12-hour period. The number of accidents follows a Poisson distribution with a parameter value of 2.

Define binomial distribution. How is it related to Poisson distribution.