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Chapter 15: Q7P (page 734)

(a) Note that (3.4) assumes P(A) is not equal to 0 since PA(B)is meaningless if P(A) = 0.

Assuming both P(A) is not equal to 0 and P(B) is not equal to 0, show that if (3.4) is true, then

P(A)=PA(B)that is if B is independent of A, then A is independent of B.

If either P(A) or P(B) is zero, then we use (3.5) to define independence.

(b) When is an event E independent of itself? When is E independent of“not E”?

Short Answer

Expert verified

(a) P(A)=PB(A)is valid.

(b) E is independent of E when role="math" localid="1664360833595" P(EE)=P(E)2and is not independent on not E

role="math" localid="1664360840497" P(EE')=P(E)×P(E')

Step by step solution

01

Definition of Independent Event

The events are said to be independent when the occurrence or non-occurrence of any event does not have any effect on the occurrence or non-occurrence of the other event.

When events are independent, apply the formula P(AB) = P(A) . P(B) where A and B are the events.

02

Important Information

Equation 3.4P(A)=PB(A)

03

Verifying the given statement

When the events A and B are independent of each other then P(AB)=P(A)×P(B).

Using the conditional probability, PB(A)=P(AB)P(B).

From the obtained relations, it is observed that P(A)=PB(A).

Thus B is independent of A and vice versa.

It can be observed that E is independent of E when P(EE)=P(E)2and is not independent on not E P(EE')=P(E)×P(E').

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Most popular questions from this chapter

(a) A weighted coin has probability of 23¯of showing heads and 13of showing tails. Find the probabilities of in two tosses of the coin. Set up the sample space and the associated probabilities. Do the probabilities add to 1 as they should? What is the probability of at least one head? What is the probability of two heads if you know there was at least one head?

(b) For the coin in (a), set up the sample space for three tosses, find the associated probabilities, and use it to answer the questions in Problem 2.12.

Suppose that Martian dice are 4-sided (tetrahedra) with points labeled 1−4. When a pair of these dice is tossed, let x be the product of the two numbers at the tops of the dice if the product is odd; otherwisex=0.

There are 3 red and 2 white balls in one box and 4 red and 5 white in the second box. You select a box at random and from it pick a ball at random. If the ball is red, what is the probability that it came from the second box?

Set up an appropriate sample space for each of Problems 1.1 to 1.10 and use it to solve the problem. Use either a uniform or non-uniform sample space or try both.

A single card is drawn at random from a shuffled deck. What is the probability that it is red? That it is the ace of hearts? That it is either a three or a five? That it is either an ace or red or both?

Define s by the equation.s2=(1/n)∑i=1n(xi−x¯)2Show that the expected valueof.s2is[(n−1)/n]σ2Hints: Write

(xi−x¯)2=[(xi−μ)−(x¯−μ)]2=(xi−μ)2−2(xi−μ)(x¯−μ)+(x¯−μ)2

Find the average value of the first term from the definition ofσ2and the average value of the third term from Problem 2. To find the average value of the middle term write

(x¯−μ)=(x1+x2+⋯+xnn−μ)=1n[(x1−μ)+(x2−μ)+⋯+(xn−μ)]

Show by Problemthat

E[(xi−μ)(xj−μ)]=E(xi−μ)E(xj−μ)=0   fori≠j

andevaluate6.14 (same as the first term). Collect terms to find

E(s2)=n−1nσ2
See all solutions

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