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Chapter 2: Q. 2.9 (page 53)

Suppose that an experiment is performed ntimes. For any event Eof the sample space, let n(E)denote the number of times that event Eoccurs and definef(E)=n(E)/n. Show that f(·)satisfies Axioms1,2,and3.

Short Answer

Expert verified

Therefore,

f(·) satisfies Axioms 1, 2, and 3.

Hence Proved.

Step by step solution

01

Given information.

letn(E)denote the number of times that event Eoccurs and definef(E)=n(E)/n.

02

Explanation.

f(E)=n(E)/n. Axiom1. By definitionn(·)is a counting function. Therefore, localid="1649236178436" f(·)positive. Axiom2. f(Ω)=n(Ω)/n=1( Every time we make the experiment some output is obtained, that is,localid="1649236192535" Ωhappened) Axiom3. Let Emm=1∞be a numerable collection of mutually exclusive events (E∩Ej=∅∀i≠j), Then:

f∪m=1∞Em=n∪m=1∞Em/n

=(*)∑m=1∞nEm/n=∑m=1∞fEm,

where the equality (*)holds by the mutually exclusive condition ofEmm=1∞.

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Most popular questions from this chapter

The game of craps is played as follows: A player rolls two dice. If the sum of the dice is either a2,3,or12, the player loses; if the sum is either a 7or an 11, the player wins. If the outcome is anything else, the player continues to roll the dice until she rolls either the initial outcome or a 7. If the 7comes first, the player loses, whereas if the initial outcome reoccurs before the 7appears, the player wins. Compute the probability of a player winning at craps.

Hint: Let Eidenote the event that the initial outcome is iand the player wins. The desired probability is ∑i=1212P(Ei). To compute P(Ei), define the events Ei,nto be the event that the initial sum is i and the player wins on the nth roll. Argue that

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