课程名称︰机率 课程性质︰必修 课程教师︰林守德 开课学院：电资学院 开课系所︰资讯系 考试日期（年月日）︰2012/4/2 考试时限（分钟）：3小时 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Probability 2012 Midterm (Prof. Shou-de Lin) 4/2/12 14:30-17:30pm Total Points: 120 You can answer in either Chinese or English. 1. Please draw the shapes of the following histograms:(a) Symmetric unimodal (b) Bimodal (c) Positively skewed. (6 pts) 2. X, Y, and Z are three events. Can you propose a real-world example of them that satisfy both of the following conditions (7 pts): (a) X and Y are independent. (b) X and Y become dependent given Z. 3. Given Baye's rule: P(B|A) = P(A|B)*P(B)/P(A). Which term is called the prior probability? Which term is the Likelihood probability? Which term is the posterior probability? (6 pts) 4. You are given two red envelops, and told that one contains X dollars and the other contains 2X. After one of the envelop is picked, you are given a chance to swap. Assuming the current envelop has Y dollars. if you swap, there is 1/2 chance you would obtain 2Y and 1/2 chance 0.5Y, therefore the expectation value of swapping is 1/2(2Y+0.5Y)=1.25Y. if you don't swap, the expectation value is Y. So you HAVE to swap anyway. Is this statement correct? If not, can you explan why by using the axioms of probability(i.e. random experiment, outcome, event, etc)? (8 pts) 5. Suppose there are two companies A and B producing environment detectors. Company A claims that there is 95% chance their detector alarms when environment is abnormal. Company B claims that if their detector alarms, there is 50% chance the environment is abnormal. Suppose the probability of abnormal environment is 1%. Given the current information, is it possible to determine which company produces a better alarm? Explain you answer. (6 pts) 6. You got a binary image, but found that it's too dark to read. You then try to modify its lightness. Suppose the mean and variance of original image is 50 and 16 respectively in light. You want to change it to 128 and 64 respectively by linear transformation(Y=aX+b). How to determine a and b for this transformation? (8 pts) 7. if E(X^r)=3^r, r = 1, 2, 3..., find the moment-grnerating function M(t) of X and the p.m.f. of X. (8 pts) 8. Typist A types 100 words per minute and makes on average 3 errors per minute; typist B types 150 words per minute and makes on average 4.2 errors per minute, both following Poisson distribution for making errors. When an article comes in, we flip a fair coin to determine who should be the typist. Suppose there is a typed article of 300 words without any error, what is the likelihood it is typed by A? (8 pts) 9. Let two independent random variables X follow Gamma(α1, θ) and Y follow Gamma(α2, θ). What is the distribution of X+Y? (5 pts) 10. Given the following random experiments, please comment whether each of them is likely to produce a random variable that follows a Poisson distribution, and explain why: (9 pts) 1. Observing the number of people entering CSIE R104 front door from 14:20 -15:00 every Mon. 2. Observing the number of cars passing 长兴街警卫亭 every Monday from 10- 11am. 3. Observing the number of cars passing 新生南路忠孝东路交叉口 at 5-6pm ever Mon. 11. If 10 observations are taken independently from a chi-square distribution with 19 degrees of freedom, find the probability that exactly 2 of the 10 sample items exceed 30.14. (9 pts) 12. Please describe the relationship between Poisson, Exponential, Gamma, and Chi-square distributions. (5 pts) 13. in NTU CSIE department, there is a probability course with 3 classes, each has 50 students. In NxxU CSIE department, they also offer three probability classes, the chair of NxxU CSIE then claims that their "teaching quality" is better then NTU CSIE since their class has on the average fewer (33.3 vs 50) students. Do you agree with this statement? Why? (7 pts) 14. (a) There are three boxes: a box containing two gold coins, a box with two silver coins, and a box with one of each. After choosing a box at random and withdrawing one coin at random, you find that the chosen coin is a gold coin. What is the probability that the remaining coin is also gold. (5 pts) (b) For the same three boxes: After choosing a box at random, a person looks at the box and then intentionally reveal one gold coin. Now what is the probability that the remaining coin is also gold. (5 pts) 15. The mean of a Poisson random variable X is 1. Compute P(1-2σ<X<1+2σ), where σ is the STD. (6 pts) 16. Show that the geometric distribution is memoryless. (12 pts) Appendix: Poisson Distribution f(x) = (λ^x)*(e^-λ)/x!, μ=λ, σ^2=λ. Exponential Distribution f(x) = λ(e^-λx), let θ=1/λ, σ^2=θ^2. Gamma Distribution ∞ f(x) = (λ^α)*(x^(α-1))*(e^-λx)/Γ(α), where Γ(t) = ∫ y^(t-1)*e^-y dy, 0 let θ=1/λ, M(t)=(1-θt)^-α, μ=αθ, σ^2=α*θ^2. Chi-square Distribution f(x) = x^(r/2-1)*e^(-x/2)/Γ(r/2)*x^(r/2), 0≦x＜∞, μ=αθ=(r/2)*2=r, σ^2=α*θ^2=(r/2)*2^2=2r. Chi-square table: (略，查表用) --

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