(10) 4 Answer the following (5) a) (5) b) Differentiate between descriptive and inferential statistics with an example for each State the Central Limit Theorem (CLT) The formal statement is the most important theorem of probability and concept of statistical inference Why Explain your answer using graphs to illustrate your point Markov inequality For a random variable X 0 with mean 0, and any number t 0 P(X4) Note that the Markov inequality is restricted to non negative random variables Chebyshev inequality For a random variable X with (finite) mean and variance 2, and for any number t 0, P(X ) Remark When Markov inequality is applied to (X )2, we obtain Chebyshev's inequality Markov inequality is also used in the proof of Hoeffding's inequality Hoeffding versus Chebyshev 4 points possible (graded) Let X1, X2, , X, id Unif (0,6) benzi i d uniform random variables on the interval 0,6 for some positive b Suppose is small (i e n 30) so that the central limit theorem is not justified Find an upper bound on the probability that the sample mean is far away from the expectation (the true mean) of X More specifically, find the respective upper bounds given by the Chebyshev and Hoeffding inequalities on the following probability P for c 2 and c 6 Each answer is numerical Using Chebyshev inequality P(X EX2) P(X EX 6) Using Hoeffding inequality P(X EX2) P (X EX 6) where VarXi Consider a Markov chain Xn, n 0,1, on the state space S 0, 1, 2 Suppose that the Markov chain has the transition matrix 4 10 10 10 4 2 P 3 4 3 10 10 10 2 4 4 10 10 10 1 Show that the Markov chain has a unique stationary mass 2 Let h denote the stationary mass of the Markov chain Find h(x) for all x S 3 Show that the Markov chain has the steady state mass 4 Let h denote the steady state mass of the Markov chain Find h (x) for all x S Explain why assessing the normality of a variable is often important Choose the correct answer below A Determining whether a variable is normally distributed is only important for samples that are less than 30 OB If the data is not normally distributed, there is no statistical calculations or statistical inference that can be done C Many procedures and analysis in statistical inference require the data to be normally distributed to be valid

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Question: (10) 4. Answer the following: (5) a) (5) b) Differentiate between descriptive and inferential statistics with an example for each. State the Central Limit

(10) 4. Answer the following: (5) a) (5) b) Differentiate between descriptive and inferential statistics with an example for each. State the Central Limit Theorem (CLT). The formal statement is the most important theorem of probability and concept of statistical inference. Why? Explain your answer using graphs to illustrate your point. Markov inequality For a random variable X>0 with mean > 0, and any number t > 0: P(X4) Note that the Markov inequality is restricted to non-negative random variables. Chebyshev inequality For a random variable X with (finite) mean and variance 2, and for any number t 0, P(X->) Remark: When Markov inequality is applied to (X-)2, we obtain Chebyshev's inequality. Markov inequality is also used in the proof of Hoeffding's inequality. Hoeffding versus Chebyshev 4 points possible (graded) Let X1, X2,..., X, id Unif (0,6) benzi.i.d. uniform random variables on the interval [0,6] for some positive b. Suppose is small (i.e. n

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