Question

Define a random variable X to be stochastically greater than a random variable Y if FX(t) < Fy(t) for all t and FX(t) < Fy(t) for some t. Let X denote the toss on which the first head appears in repeated tosses of a fair coin, and let y denote the role on which the first 'l' appears in repeated rolls of a fair coin. Which of these two random variables is stochastically greater than the other? (See exercise 1.49 in C&B.)

Answer 1

Solution:- Let, :) Before this problem, We need to understand the geometric distribution + Let X ~ Geometric (GCP)) - real random Variables, x has distribution with real parameter p satisfying 02P21. The Geometric random Variable has mass function, f(x) = PCI-P)? x=0,1,2 .--. - The Geometric distribution can be used to model, the no of failures before the first Success en repeated, mutually independent Bernoulli trials. The cumulative distribution function of the support of x is an F(x) = P(x2x) =1-(1-P)?#! x30,1,2---- Now, based on the above in formation, x can be modelled as a Geometric random Variable with parameter palla

Similarly, y can be modelled as a Geometric random Vociable with Parameter P-118 Ex(x) = 1-(12) **1 X=1 1=0,1,2 ... Fy(x) = 1 (46)%*' 23 8 Ex(x) Fu(x) 0 · 516 - 4 1 0.9722 6.875 0.9953 0.9375 0.9992 6196875 0.9998 y 0.984375 0.9999 5 using the table, We can say that 1 x (2) < Ev (2) 4x=0,1,2 - .. .. The random variable x greater than y. is stochostically