You are to show your work in solving these problems.
You are to show your work in solving these problems. You can use symbols, Rcode, or written description. The problems can be written out by hand, or you may use a text editor like MS-WORD. Your solutions should be numbered.
Please answer the following questions.
1) What would be the most appropriate probability distribution for each of the following random variables? Please justify your answers.
a) Whether a tumor is benign or malignant.
b) Number of people with a malignant tumor out of 10 patients with tumor.
c) Size of tumors.
d) Number of people diagnosed with malignant tumors in California every year.
2) Consider the Binomial distribution with n=10 and p=0.3. In other words, let X ~ Binomial(10, 0.30). Do the following tasks:
a) Plot the probability distribution (i.e. the probability mass function).
b) Plot the cumulative distribution function.
c) Write down the mean and standard deviation for the distribution.
d) What is the probability that the value of the random variable is 2? In other words, Prob[X=2]
e) Find the lower tail probability of 4. In other words, find Prob[X<=4].
f) What is the probability that the value of the random variable is greater than 2 and less than or equal to 4? In other words, Prob[X>2 and X<=4]. This is a little tricky – pay close to the symbols.
3) Suppose patients suffering from major trauma like a gun shot wound or automobile accident arrive at a metropolitan emergency room (ER) at a rate of 1 per hour on average. What is the probability that during the next hour:
a) No major trauma patients arrive at the ER?
b) Exactly 1 trauma patient arrives?
c) The ER is swamped and exceeds the ER capacity? The ER is considered swamped and exceeding capacity if 3 or more major trauma patients arrive in the same hour.
4) The National Heart, Lung, and Blood Institute defines the following categories based on Systolic Blood Pressure (SBP):
- Normal: SBP ≤ 120.
- Prehypertension: 120 < SBP ≤ 140.
- High blood pressure: SBP > 140.
From prior experience, it is known that SBP in the United States follows a normal distribution with a mean MU=125 and standard deviation SIGMA=SQRT(152)=12.329. In symbols: SBP ∼ N(125, 152).
a) If a US person is randomly selected, what is the probability the person would be considered Normal for SBP? What is the probability the person would be considered Prehypertensive? What is the probability the person would have high blood pressure?
b) Find the interval for SBP that includes the middle 68% of the population? Similarly, find the intervals for SBP for the middle 95% and the middle 99.7% of the population. What happens to the intervals as you include more of the population?
c) What is the lower tail probability for SBP equal to 115? What is the upper tail probability for SBP equal to 115?
5.) Assume that the Body Mass Index (BMI) in United States follows a Normal Distribution with mean 27 and standard deviation of SQRT(62)=7.87. In other words, BMI ~ N(27, 62). Following the recommendation by National Heart, Lung, and Blood Institute, we define the following BMI categories:
- Underweight: BMI ≤ 18.5.
- Normal weight: 18.5 < BMI ≤ 25.
- Overweight: 25 < BMI ≤ 30.
- Obesity: BMI > 30.
a) What is the probability of each group?
b) What is the probability of being either underweight OR obese?
c) What value of BMI would you have to have to be in the 95 percentile of the US population?
d) If your BMI score is 24.5, what is your percentile relative to the US population?
e) Denote BMI as X. Find the value x such that P(X ≤ x) = 0.2. Next, find the value x such that P(X >x) = 0.2. Interpret the results.
6) For the probability distributions Binomial(100, 0.3) and N(30, 21), find the lower tail probability of 35 and the upper tail probability of 27 (Please be careful with the variance / standard deviation conversion for the normal distribution). Compare the results based on the two distributions.
Congratulations! You’ve completed the third Assignment. Please save your R-code, because you can re-use or cannibalize this code in future assignments.
Your write-up should address each question. You should label each answer by question number. Please use the following file naming convention: Assign3_yourlastname.PDF. Your write up must be in PDF format. Please note, MS-WORD and most text processors allow you to save your documents in PDF format
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