(a) the probability that at the next tossing of the coin heads will show；

(b) a posterior probability of 0.3 ≤ X ≤ 0.7, given the coin is tossed 10 times and heads shows 6 times?

2. (10%) The random variables X and Y are independent and each is uniformly distributed in the interval (0, a). Find and plot the density of the random variable Z = |X - Y| ?

(a) the sampling distribution of

(b) the mean and standard deviation of the distribution of

4. (10%) Suppose that a system contains a certain type of component whose time (in years) to failure is given by the random variable T, distributed exponentially with mean time to failure 5 years. If 50 of these components are installed in different systems, what is the probability that at least 5 of them are still functioning at the end of 8 years?

(a) Find the maximum likelihood estimation (MLE) of λ, based on a random sample size n?

(b) Describe a practical situation in which one would suspect that the shift exponential distribution is a plausible model?

6. (8%) Consider the following confidence interval for μ with known standard deviation σ: where α1 + α2 = α. Now we usually find the interval for α1 = α2 = α/2, not the ones for α1 ≠ α2. Show if there is any advantage to a "symmetric" confidence interval?

7. (12%) To test the hypothesis that a coin is fair (i.e., p = 0.5) by a number of tosses of the coin, we wish to impose the following restrictions:(1) The probability of rejecting the hypothesis when it is actually correct must be 0.05 at most.(2) The probability of accepting the hypothesis when actually p differs from 0.5 by 0.1 or more(i.e., p ≥ 0.6 or p ≤ 0.4) must be 0.05 at most.Determine the minimum sample size that is necessary, and state the resulting decision rule?

8. (10%) An equipment manufacturer has developed a new electrical product. He claims that the product has a mean life time of 10,000 hours, with a standard deviation of 150 hours. A customer tested 16 random samples of the product and obtained a mean life time of 9,775 hours.Evaluate at the 0.05 level of significance whether the life time announcement for the new product is reliable?

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