Introductory Probability

AMA1006 Basic Statistics, AMA1104 Introductory Probability Assignment2Due at 17:00, March 24, 2016, Friday. Please submit your homework into the assignment box at ?oor 7 or 8, core T. The assignment box will be available for about one week ahead of the deadline, and with thelecturer’snameprinted on the front. The markers of the assignments are research students from department AMA. Please show your courtesy to them by boxing your ?nal solutions, like:
…therefore the desired probability is P(A) = 0.334 .
Full marks: 100.
1. (20 marks) Let X be a discrete random variable of which the probability function f is de?ned by the following table.
x -3 -1 0 2 3 f(x) 0.1 0.2 0.3 0.2 a
a. Find a. b. Compute the mean and variance of X. c. Compute the mean and variance of X -3. d. Compute the mean and variance of X2.
2. (20 marks) a. Let X ~ Binomial(10,0.2). Find P(X = 0) and P(X > 2). b. Let X ~ Binomial(10000,0.0005). Use Poisson approximation to evaluate P(X < 3). c. Let X ~ Poisson(2). Find P(X > 2). d. Let X ~ Poisson(2). Find the conditional probability P(X > 3|X = 2). 3. (10 marks) The average life of an engine manufactured by a car company is 10 years, with a standarddeviationof2years. Assumethatthelivesoftheenginesfollowanormaldistribution. Find the probability that the engine has a life between 7.5 years and 13 years.
4. (20 marks)
a. Abox contains 10ballsofwhich 5arered. Suppose wepick out5 ballswithoutreplacement. What is the probability that among these 5 balls picked out, 3 are red? b. A box contains 10 balls of which 5 are red. Suppose in each round of an experiment we pick out a ball randomly from the box, take record of its color, and then put it back. We do this experiment repeatedly until we get 3 red balls. What is the probability that we have already done the experiment 6 rounds in total?
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5. (20 marks) A die is unfair and when tossed, it gives “1” with probability 0.001, and “2” with probability 0.4.
a. If we toss the above said die ?ve times, what is the probability that we get at least one face “2”? b. Suppose we toss the above said die one hundred times. Estimate the probability that we get no more than 30 times the face “2”. c. Supposewetoss800timestheabovesaiddie. Estimatetheprobabilitythatwegetexactly two times the face “1”.
6. (10 marks) a. Let X ~ Uniform(2,5). Find P(X > 4), P(X = 4),E[X], and Var(X). b. LetW bearandomvariablethathasexponentialdistributionwithmean3.2. FindP(X > 3) and P(X > 5 + 3|X > 5).
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