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Chapter 15: Q6P (page 749)

A card is drawn from a shuffled deck. Let x=10if it is an ace or a face card; x=−1if it is a2 ; andx=0 otherwise.

Short Answer

Expert verified

The required values are mentioned below.

μ=3var(x)=28413σ=4.67

Step by step solution

01

Given Information

A card is drawn from a shuffled deck.

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=∑xiPiμ=10×1652−1×452μ=3

The variance is given below.

var(x)=∑xi−μ2pxivar(x)=10-32×1652+-1-32×452+32×3252var(x)=28413

The standard deviation is given below.

σ=var(x)σ=28413σ=4.67

Thegraph is shown below.

Hence, the required values are mentioned below.

μ=3var(x)=28413σ=4.67

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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 .

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

A letter is selected at random from the alphabet. What is the probability that it is one of the letters in the word “probability?” What is the probability that it occurs in the first half of the alphabet? What is the probability that it is a letter after x?

(a) Find the probability that in two tosses of a coin, one is heads and one tails. That in six tosses of a die, all six of the faces show up. That in12tosses of a12 -sided die, all12 faces show up. That in n tosses of an n-sided die, all n faces show up.

(b) The last problem in part (a) is equivalent to finding the probability that, when n balls are distributed at random into n boxes, each box contains exactly one ball. Show that for large n, this is approximatelye−n2πn.

Computer plot on the same axes the normal density functions with μ=0σ=1and, 2, and 5. Label each curve with itsσ.

A so-called 7-way lamp has three 60-watt bulbs which may be turned on one or two or all three at a time, and a large bulb which may be turned to 100 watts, 200 watts or300 watts. How many different light intensities can the lamp be set to give if the completely off position is not included? (The answer is not 7.)
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