calculate the pressure of 1 mole of methane as a function of container volume at 0°C (273 K) at suitable volumes from 22.4 L to 0.05 L.

The van der Waals equation is an equation of state that applies to real gases. For 1 mole of a gas, the van der Waals equation is

where R is the gas constant (0.0821 L atm K-‘ mol-‘) and T is the Kelvin temperature. The constants a and b are constants particular to a given gas, and correct for the attractive forces between gas molecules, and for the volume occupied by the gas molecules, respectively. For methane (CK), the constants are a = 2.253 L2 atm and b = 4.278 x L. Using the rearranged form of the van der Waals equation

calculate the pressure of 1 mole of methane as a function of container volume at 0°C (273 K) at suitable volumes from 22.4 L to 0.05 L. Use one of the custom functions described in this chapter to calculate the first and second derivatives of the P-V relationship. Compare with the exact expressions

Compute the information gain of the term “elections” according to Eq. 5.7.

Consider the term “elections” which is present in only 50 documents in a corpus of 1000 documents. Furthermore, assume that the corpus contains 100 documents belonging to the Politics category,….

Predict the probabilities of categories Cat and Car of Test2 on the toy corpus example in Sect. 5.3.5.2. You can use the multinomial na¨ıve Bayes model with the same level of smoothing as used in the example in the book. Return normalized probabilities that sum to 1 over the two categories. 2.       Na¨ıve Bayes is a generative model in which each class corresponds to one mixture component. Design a fully supervised generalization of the nai¨ıve Bayes model in which each of the k classes contains exactly b > 1 mixture components for a total of b · k mixture components. How would you perform parameter estimation in this model?

1.       Predict the probabilities of categories Cat and Car of Test2 on the toy corpus example in Sect. 5.3.5.2. You can use the multinomial na¨ıve Bayes model with the same….

What are the possible advantages and disadvantages of using such an approach?

1.       Na¨ıve Bayes is a generative model in which each class corresponds to one mixture component. Design a semi-supervised generalization of the nai¨ıve Bayes model in which each of the….