Exercise 1 . Let *1, X2 . and /1, 12 . be sequences of random variables such that * * * and gt; gt; gt;' both in probability . Prove * * * * Y in...

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Exercise 1 . Let *1, X2 . ... and /1, 12 . ... be sequences of random variables such that* * * and > > >' both in probability . Prove * * * * Y in probability ."Exercise 2 ( Normal numbers , or a monkey at a typewriter ) . Define a number a E ( 0 , 1]to be simply normal in base 10 if for every digit KE ( 0 , 1 , 2 . .... If we have*lim .# digits equal to k amongst first ~ digits*112 -1 0 010'Use the Strong Law of Large Numbers to show that a real number chosen uniformly in10 , 1 ] will almost surely be simply normal in base 10 .*Define a number a { \O , I) to be simply normal if it is simply normal in every base*be ( 2 , 3 , 4 , ... J. Show that a real number chosen uniformly in\O , I will almost surely*be simply normal .Connect the above ideas to the" monkey at a typewriter paradox " : with probability*I , a monkey typing at a typewriter will type out the script of your favorite Shakespeare*play , with the right frequency .*Exercise 3 ( Probabilities of events as frequencies of occurrence ) . Use the Strong Lawor the Weak Law to interpret the probability of an event A as the frequency with whichit occurs in an infinitely long sequence of independent trials .Exercise 1 . Suppose that *1, X2 . ... are i. id. , YI, Y2 . ... are i. id., and that Yit ... + Y `is always nonzero ( or almost surely nonzero ) . Assume that all random variables have*finite variance ( and hence finite mean too ) . Prove thatXIt ... + X x\[[X]LIA ,Vit ... + Y ~`almost surely .Exercise 5 . Suppose that *1 * *2 . ... are i. id ., and that all random variables are*positive ( or positive almost surely ) . Prove that*( * 1 ... In ) } / { } @ Ellos (* )almost surely*.Exercise 6 ( A version of Benford's law ) . Let dec ( * ) and sig ( * ) denote the decimalpart and significant digit of a positive real number * . For example if * = 689. 737 , thendec (*) = 0. 737 and sig ( * ) = 6 .