05 (15 points) The Colorall Paint factory uses "agent A" in the paint manufacturing process. There is an average of 3.45 particles of agent A in a cubic metre of the air emitted during the production process. (a) Let X = the number of particles of agent A in a cubic metre of the air emitted. What is the probability distribution of X? (b) What is the probability that there will be five particles of agent A in a cubic metre of air emitted from the factory? (c) What is the probability that there will be at least four particles of agent A in a cubic metre of air emitted from the factory? (d) Find the expected value, variance and the standard deviation of X. Q1 (10 points) A survey was conducted to study the proportion of smokers in UK. The question, "Do you smoke?" was asked to 10000 randomly selected UK residents. Results showed that 47.5% of them smoke. Determine: (a) Population (b) Sample (0) Parameter (d) Statistic (e) Variable Q3 (20 points) The medical records of the 445 male diabetics patients reporting to a clinic during one year provide the following results: Diabetes in Parents Age of the Patient I Yes I No Below 20 l 50 l 48 20 to 40 | 72 | 30 Above 40 103 92 Find the probability that: (a) The age of the patients is above 40 and their parents are not diabetics. (b) The age of the patients is above 40 or their parents are not diabetics. (c) The age of the patients is above 40 given that their parents are not diabetics. Q6 (15 points) The literacy rate for a nation measures the proportion of people age 15 and over who can read and write. The literacy rate for women in Afghanistan is 15%. Let X = the number of Afghani women you ask until one says that she is literate. (a) What is the probability distribution of X? (b) What is the probability that you ask six women before one says she is literate? (c) What is the probability that you must ask less than 4 women? (d) Find the mean, variance and standard deviation of X. 02 (25 points) The number of days the first 22 heart transplant patients at Stanford survived after their operation were: 3, 15, 20, 23, 26, 32, 32, 32, 46, 48, 48, 56, 64, 68, 69, 69, 70, 71, 71, 80, 85, 125 (a) Find the median. (b) Find the 90th percentile. (c) Find the first and the third quartile. (d) Is there any outlier in this data set? (e) Construct a box plot. Q4 (15 points) According to the Mendelian theory of inherited characteristics, a cross fertilization of related species of red- and white- flowered plants produces a generation whose offspring contain 25% red-flowered plants. Suppose that a horticulturist wishes to cross 7 pairs of the cross-fertilized species. Of the resulting 5 offspring, what is the probability that (a) Let X = the number of red- flowered plants. What is the probability distribution of X? (b) There will be no red-flowered plant. (c) There will be 4 or more red- flowered plants. (d) Find the expected value, variance and the standard deviation of X. (d) The age of the patients is between 20 and 40 or their parents are diabetics. If A="The age of the patient is below 20" and B="Patient's parents are diabetics" (e) Are events A and B independent? Justify tour answer. (f ) Are events A and B mutually exclusive? Justify tour