PMFs

Section 3.1 PMFs

A Pmf object is like a FreqTab that contains probabilities instead of frequencies. So one way to make a Pmf is to start with a FreqTab. For example, here’s a FreqTab that represents the distribution of values in a short sequence.

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Listing 3.1.1. Python Code

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$ from empiricaldist import FreqTab

ftab = FreqTab.from_seq([1, 2, 2, 3, 5])
ftab
1    1
2    2
3    1
5    1
Name: , dtype: int64

The sum of the frequencies is the size of the original sequence.

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Listing 3.1.2. Python Code

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$ n = ftab.sum()
n
np.int64(5)

If we divide the frequencies by n, they represent proportions, rather than counts.

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Listing 3.1.3. Python Code

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$ pmf = ftab / n
pmf
1    0.2
2    0.4
3    0.2
5    0.2
Name: , dtype: float64

This result indicates that 20% of the values in the sequence are 1, 40% are 2, and so on.

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We can also think of these proportions as probabilities in the following sense: if we choose a random value from the original sequence, the probability we choose the value 1 is 0.2, the probability we choose the value 2 is 0.4, and so on.

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Because we divided through by n, the sum of the probabilities is 1, which means that this distribution is normalized.

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Listing 3.1.4. Python Code

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$ pmf.sum()
np.float64(1)

A normalized FreqTab object represents a probability mass function (PMF), so-called because probabilities associated with discrete values are also called "probability masses".

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The empiricaldist library provides a Pmf object that represents a probability mass function, so instead of creating a FreqTab object and then normalizing it, we can create a Pmf object directly.

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Listing 3.1.5. Python Code

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$ from empiricaldist import Pmf

pmf = Pmf.from_seq([1, 2, 2, 3, 5])
pmf
1    0.2
2    0.4
3    0.2
5    0.2
Name: , dtype: float64

The Pmf is normalized so the total probability is 1.

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Listing 3.1.6. Python Code

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$ pmf.sum()
np.float64(0.8500000000000001)

Pmf and FreqTab objects are similar in many ways. To look up the probability associated with a value, we can use the bracket operator.

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Listing 3.1.7. Python Code

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$ pmf[2]
np.float64(0.4)

Or use parentheses to call the Pmf like a function.

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Listing 3.1.8. Python Code

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$ pmf(2)
np.float64(0.4)

To assign a probability to a value, you have to use the bracket operator.

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Listing 3.1.9. Python Code

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$ pmf[2] = 0.2
pmf(2)
np.float64(0.2)

You can modify an existing Pmf by incrementing the probability associated with a value:

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Listing 3.1.10. Python Code

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$ pmf[2] += 0.3
pmf[2]
np.float64(0.5)

Or you can multiply a probability by a factor:

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Listing 3.1.11. Python Code

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$ pmf[2] *= 0.5
pmf[2]
np.float64(0.25)

If you modify a Pmf, the result may not be normalized — that is, the probabilities may no longer add up to 1.

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Listing 3.1.12. Python Code

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$ pmf.sum()
np.float64(0.8500000000000001)

The normalize method renormalizes the Pmf by dividing through by the sum — and returning the sum.

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Listing 3.1.13. Python Code

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$ pmf.normalize()
np.float64(0.8500000000000001)

Pmf objects provide a copy method so you can make and modify a copy without affecting the original.

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Listing 3.1.14. Python Code

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$ pmf.copy()
1    0.235294
2    0.294118
3    0.235294
5    0.235294
Name: , dtype: float64

Like a FreqTab object, a Pmf object has a qs attribute that accesses the quantities and a ps attribute that accesses the probabilities. It also has a bar method that plots the Pmf as a bar graph and a plot method that plots it as a line graph.

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