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5. Minimum and Average In this exercise you will discover properties of the sample minimum and the sample average of i.i.d. samples taken from two well-known distributions. If necessary, you can leave answers in terms of the standard normal cdf . (a) Let U1,U2,,Un be i.i.d. uniform (0,1) random variables and let Xn=min{U1,U2,,Un}. Find the survival function of Xn and hence find the density of Xn. (b) Let T1,T2,,Tn be i.i.d. exponential () random variables and let Yn=min{T1,T2,,Tn }. Find the survival function of Yn and hence identify the distribution of Yn as one of the famous ones; remember to provide the relevant parameters. (c) Let U1,U2,,Un be i.i.d. uniform (0,1) random variables and let Vn=n1i=1nUi. For large n, find an approximation to the survival function of Vn. (d) Let T1,T2,,Tn be i.i.d. exponential () random variables and let Wn=n1i=1nTi. For large n , find an approximation to the survival function of Wn..

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5. Minimum and Average In this exercise you will discover properties of the sample minimum and the sample average of i.i.d. samples taken from two well-known distributions. If necessary, you can leave answers in terms of the standard normal cdf . (a) Let U1,U2,,Un be i.i.d. uniform (0,1) random variables and let Xn=min{U1,U2,,Un}. Find the survival function of Xn and hence find the density of Xn. (b) Let T1,T2,,Tn be i.i.d. exponential () random variables and let Yn=min{T1,T2,,Tn }. Find the survival function of Yn and hence identify the distribution of Yn as one of the famous ones; remember to provide the relevant parameters. (c) Let U1,U2,,Un be i.i.d. uniform (0,1) random variables and let Vn=n1i=1nUi. For large n, find an approximation to the survival function of Vn. (d) Let T1,T2,,Tn be i.i.d. exponential () random variables and let Wn=n1i=1nTi. For large n , find an approximation to the survival function of Wn.

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