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Accuracy of MSI testing in predicting germline mutations of by Chen S.

By Chen S.

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0( > 1) defines the distribution function of a continuous random variable X in the range (0 ::;; X ::;; 1). Determine the first four moments of the distribution of X, and hence calculate the coefficients of skewness and kurtosis. What happens when a = 1? 9 A continuous random variable X has, for X = x, the probability density function proportional to e- X (1 +X)2 in the range (-1 ::;; X < co), and zero otherwise. /3 and 2 respectively. Also, show that m, the median of the distribution, satisfies the relation e(m+l) = 1+(m+2)2.

If xli) are the proportions of Sj Sj in the nth generation, find their values explicitly, and hence prove that as n ~ 00, these proportions all tend to the limiting value 2/k(k-1). Hence, or otherwise, verify that for k = 4, (i "# j "# 1"# t). 97 In the simplest type of weather forecasting-"rain" or "no rain" in the next 24 hours-suppose the probability of raining is p (>t), and that a forecaster scores a point if his forecast proves correct and zero otherwise. In making n independent forecasts of this type, a forecaster, who has no genuine ability, predicts "rain" with probability A.

X r I 2r r=1 where Verify that this series expansion for the distribution function of X is asymp· totic by proving that, for any n, the remainder is less in absolute value than the last term taken into account. ~-. 1+ 2' 2, " y 2n x ,= 1 • X • r . 1 (x 2 + 2k). 40 A random variable X has the probability density function J(X = x) = ksin 2n xcosx defined in the range (-nI2 ::; X ::; nI2), where k is a function of the positive integer II. Determine k and calculate the probability that (i) X ::; () and IXI ~ () for any given (); and n (ii) -6::; X :e;; 1t 6' Also, prove that where 12n+ 1 satisfies the recurrence relation (2n+l)212n+l = 1+(2n)(2n+l)1 2n _ 1, with II = 1.

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