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By Van Der Merwe A. J., Du Plessis J. L.

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Extra info for A Bayesian Approach to Selection and Ranking Procedures: The Unequal Variance Case

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F. f. F . f. s in a useful manner ([54, p. 307, Theorem 9]). v. with E(Y ) = 0 and Var(Y ) = σY2 φY (t) = 1 − σY2 t2 /2 + K(t) where lim K(t)/t2 = 0. t→0 Now we can state the central limit theorem. 3 • • • • • • • • If X, X1 , X2 , . . d. s; µ = E(X); σ 2 = Var (X); S(n) = X1 + X2 + . . f. of Z(n); Z ∼ N (0, 1); t 1 FZ (t) = √ exp(−u2 /2) du 2π −∞ ws-tb-9x6 September 17, 2013 14:11 8946-Problems in Probability(2nd Edition) Problems ws-tb-9x6 55 then (∀t ∈ ℜ)( lim Fn (t) = FZ (t)). n→∞ We now prove the CLT as follows.

Then we say that Y has a log normal distribution. (a) Why do we use the name “log-normal”? (b) Find an expression for fY (t). (c) Sketch the graph of fY . (11) Let X ∼ N (0, 1). 3413. Verify this using numerical integration. (12) Buffon’s needle. Suppose that a plane is ruled by equidistant parallel lines at distance 1 apart: for example represent the lines by y = n(n = 0, ±1, ±2, ±3, . ). A needle of length L < 1 is thrown randomly on the plane. (a) Prove that the probability that the needle will intersect one of the parallel lines is 2L/π.

B) Carefully describe a practical situation which would give rise to a random variable whose distribution you would expect to be a Normal distribution. State some sensible values of the parameters µ and σ involved. (c) Carefully describe a practical situation which would give rise to a random variable whose distribution you would expect to be a non-symmetric or skewed distribution. (5) In many elementary statistics books, we find a statement of the socalled “empirical rule”. 95. The aim of this exercise is to examine the limitations of this “rule”.

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