Answer:

1. Imports & setup

2. Generate two Binomial arrays (10000 samples each)

3. Case (a): Empirical histogram vs analytical PMF(n=100, p=0.3)

Note for Case (b)

For n=10000, plotting the full support k=0,…,10000 is visually uninformative because almost all probability mass is concentrated near the mean np=6000. Therefore, the plot for Case (b) displays only the range of k values observed in the 10000 samples, which provides a clearer comparison between the empirical histogram and the analytical PMF.

4. Case (b): Empirical histogram vs analytical PMF (n=10000, p=0.6)

5. Brief numerical check (empirical mean/variance vs theoretical)

Conclusion

In both cases, the empirical histograms from 10000 simulated samples closely match the analytical PMFs computed with scipy.stats.binom. This confirms that the NumPy-generated samples follow the Binomial distribution. Case (b) is much more concentrated around np due to the large n, so the observed spread of k values is relatively narrow compared with the full support.

a) Which data process, (P_X) or (P_Y), has a greater level (Definition 4.4)? Why?

Answer: They have the same level, typically Level 1.

Reason:By Definition 4.4, Level 1 processes are symbolic processes involving data directly obtained from physical reality.Both (X) (global average temperature increase) and (Y) (temperature observed in city A) originate from physical temperature measurements. Even though (X) is an average and (Y) is local, both are still derived from direct physical measurements, hence both correspond to Level 1 data processes.

b) Let (Z) be the random variable answering: “Is global warming true?” What is the relation between (H(Z)), H(Z|X), and H(Z|Y)?

Answer (guaranteed):

H(Z∣X)≤H(Z),H(Z∣Y)≤H(Z).

Reason: From Proposition 4.2, conditioning on additional knowledge cannot increase uncertainty. Knowing (X) (or (Y)) provides information about (Z), so the conditional entropy is not larger than the unconditional entropy.

Typical (context-based) strengthening:** Since (X) is a global-average indicator more directly linked to the global-warming claim than a single-city record (Y), one would typically expect:

H(Z∣X)≤H(Z∣Y),

equivalently I (Z;X)≥I(Z;Y). (This comparison relies on the assumption that 𝑋 X is more informative about 𝑍 than 𝑌.)

c) Let (W) be the random variable answering: “How was the weather of city A during last winter?” What is the relation between H(W), H(W|X), and H(W|Y)?

Answer (guaranteed):

H(W∣X)≤H(W),H(W∣Y)≤H(W).

Reason: Again by Proposition 4.2, conditioning on knowledge cannot increase uncertainty.

Typical (context-based) strengthening: Because (W) is specifically about city A’s winter weather, it is usually more directly related to local temperature information (Y) than to the global average (X). Thus one typically expects:

H(W∣Y)≤H(W∣X), equivalently I(W;Y)≥I(W;X). (This comparison also depends on the assumed relevance/association.)

d) What can be said about the mutual information I(Z;X), I(Z;Y), I(W;X), I(W;Y), and I(X;Y)?

Using Eq. (4.3), for any variables (A,B),

I(A;B)=H(A)−H(A∣B)=H(B)−H(B∣A)≥0.

Therefore:

I(Z;X)=H(Z)−H(Z∣X)≥0,I(Z;Y)=H(Z)−H(Z∣Y)≥0,

I(W;X)=H(W)−H(W∣X)≥0,I(W;Y)=H(W)−H(W∣Y)≥0.

Typical (context-based) ordering:

(X) is a global indicator and is usually more informative for (Z) than a single-city event (Y):

I(Z;X)≥I(Z;Y).

(Y) is local and is usually more informative for (W) than the global average (X):

I(W;Y)≥I(W;X).

For (I(X;Y)):

If (X) and (Y) were independent, then (I(X;Y)=0). In realistic climate contexts, (X) (global trend) and (Y) (local temperatures) are generally not fully independent, but local variability can be large, so (I(X;Y)) may be positive yet not necessarily large.

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e) What is the problem with the argument used by negationists?

Answer:The argument uses a local, short-term observation(Y: “city A was extremely cold”) to deny a global, long-term claim(Z: “global warming is true”), which is a mismatch of scope and evidence.

Information-theoretic interpretation:A single-city cold event (Y) typically provides limited information about the global claim (Z) (i.e., (I(Z;Y)) is small compared with the information carried by global aggregated indicators such as (X)). Hence, observing (Y) does not substantially reduce the uncertainty about (Z), and it is not a sound basis to reject the global-warming hypothesis. The argument also often reflects selective reasoning (focusing on one extreme local datapoint rather than the overall process/trend).

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