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Chapter 15: Q5P (page 749)

A random variable x takes the values with probabilities 512,13,112,16.

Short Answer

Expert verified

The required values are mentioned below.

μ=1var(x)=76σ=76

Step by step solution

01

Given Information

A random variable x.

02

Definition of the cumulative distribution function

The likelihood that a comparable continuous random variable has a value less than or equal to the function's argument is the value of the function.

03

Find the values

The mean is given below.

μ=∑0×5pxiμ=0×512+1×13+2×112+3×16μ=1

The variance is given below.

var(x)=∑xi−μ2pxivar(x)=0-12×512+1-12×13+×112+3-12×16var(x)=76

The standard deviation is given below.

σ=var(x)σ=76

Thegraph is shown below.

Hence, the required values are mentioned below.

μ=1var(x)=76σ=76

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Most popular questions from this chapter

(a) In Example, 5a mathematical model is discussed which claims to give a distribution of identical balls into boxes in such a way that all distinguishable arrangements are equally probable (Bose-Einstein statistics). Prove this by showing that the probability of a distribution of N balls into n boxes (according to this model) with N1 balls in the first box, N2in the second, ··· , Nn in thenth , is1C(n−1+N,N) for any set of numbers Ni such thatNi∑i=1nNi=N.

b) Show that the model in (a) leads to Maxwell-Boltzmann statistics if the drawn card is replaced (but no extra card added) and to Fermi-Dirac statistics if the drawn card is not replaced. Hint: Calculate in each case the number of possible arrangements of the balls in the boxes. First do the problem of 4particles in 6boxes as in the example, and then do N particles in n boxes (n>N ) to get the results in Problem19 .

A true coin is tossed 104 times.

(a) Find the probability of getting exactly 5000 heads.

(b) Find the probability of between4900and 5075 heads.

Two cards are drawn from a shuffled deck. What is the probability that both are aces? If you know that at least one is an ace, what is the probability that both are aces? If you know that one is the ace of spades, what is the probability that both are aces?

Answer

Given a family of two children (assume boys and girls equally likely, that is, probability for each), what is the probability 1/2 that both are boys? That at least one is a girl? Given that at least one is a girl, what is the probability that both are girls? Given that the first two are girls, what is the probability that an expected third child will be a boy?

(a) There are 10 chairs in a row and 8 people to be seated. In how many ways can this be done?

(b) There are 10 questions on a test and you are to do 8 of them. In how many

Ways can you choose them?

(c) In part (a) what is the probability that the first two chairs in the row are vacant?

(d) In part (b), what is the probability that you omit the first two problems in the

test?

(e) Explain why the answer to parts (a) and (b) are different, but the answers to

(c) and (d) are the same.
See all solutions

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