## STAT41700

Statistics统计学 Suppose we observe a value X = 1. What is the true value of the parameter? What is the true distribution of X?

### 4.2.14 Give an example of random variables X1, X2, … such that {Xn} converges to 0 in probability, but E(Xn) = 1 for all n.

(Hint: Suppose P(Xn = n) = 1/n and P(Xn =0)= 1- 1/n.)

### 4.4.6 Let Z1, Z2,… be ii.d. with distribution Uniform[- -20, 10]. Use the central limit theorem and Table D.2 (or software) to estimate the probability P(≥ – 4470). Statistics统计学

### 5.2.11 Suppose a fair coin is tossed 10 times and the response X measured is the number of times we observe a head. Statistics统计学

(a) If you use the expected value of the response as a predictor, then what is the predic-tion of a future response X?

(b) Using Table D.6 (or a statistical package), compute a shortest interval containing at least 0.95 of the probability for X. Note that it might help to plot the probability function of X first.

(c) What region would you use to assess whether or not a value so is a possible future **value? (Hint: What are the regions of low probability for the distribution?) Assess whether or not x = 8 is plausible.**

Thus, the shortest interval is [2, 8]

(c)We use the interval calculated above to assess the probability of So being a future value. = 0.043945 and P(X > 8), according to table D.6,equals to 0.0098 + 0.001 = 0.0108 which is relatively away from the center of the distribution,so X = 8 is not plausible.

### 5.3.7 Suppose it is known that a random variable X follows one of the following dis-tributions. Statistics统计学

(a) What is the parameter space Ω?

(b) Suppose we observe a value X = 1. What is the true value of the parameter? What is the true distribution of X?

(c) What could you say about the true value of the parameter if you had observed X =2? Statistics统计学

(a) Ω= {A,B}

(b)Based on Likelihood Inference, L(A|1) > L(B|1) = 0, A is the only possible choice of the true value of parameter, is the only possible choice of the true distribution.

(c)L(A|2)= L(B|2)= , so both A and B are possible for being the true value of the parameter.

### 6.1.1 Suppose a sample of n individuals is being tested for the presence of an antibody in their blood and that the number with the antibody present is recorded. Statistics统计学

Record an appropriate statistical model for this situation when we assume that the responses from appropriate statistical model for this situation when we assume that the responses from individuals are independent. If we have a sample of 10 and record 3 positives, graph a representative likelihood function.

The statistic model is X ~ Binomial(10, θ)whereθ∈[0,1]= s is the probability of testing positive. The likelihood function is:

6.1.3 Suppose that the lifelengths (in thousands of hours) of light bulbs are distributed Exponential(0), where θ > 0 is unknown. If we observe x = 5.2 for a sample of 20 light bulbs, record a representative likelihood function. Why is it that we only need to observe the sample average to obtain a representative likelihood?

As can be seen in the function above, the representative likelihood is a function of x,and x is the sufficient statistics.

### 6.1.8 Suppose that a statistical model is comprised of two distributions given by the following table:

(a) Plot the likelihood function for each possible data value s.

(b) Find a suficient statistic that makes a reduction in the data.

Blue line stands for the likelihood function for f_{1}.

Blue line stands for the likelihood function for f_{2}.

(b)Since L(|1) = 2L( |3), both give the same likelihood ratio. T:S→{0, 1} given by T(1) = T(2) = 0,and T(3) = 1.

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