PaC Exercise 2.6

import numpy as np

Exercise 2.6

Suppose that we independently roll two standard six-sided dice. Let X₁ be the number that shows on the first die, X₂ the number on the second die, and X the sum of the numbers on the two dice.

1. What is E[X|X₁ is even]?

Solution: By linearity

$$ E[X | X_1 \text{ is even}] = E[X_1 | X_1 \text{ is even}] + E[X_2 | X_1 \text{ is even}] = \frac{2+4+6}{3} + \frac{7}{2} = \frac{15}{2} = 7.5. $$

Intuitively, this makes sense since we expect this expectation value to be slightly larger than E[X] = 7 since the even numbers are larger on average than the odd numbers on a standard die.

2. What is E[X|X₁ = X₂]?

Note that if X₁ = X₂ then X is even. Thus,

$$ E[X | X_1 = X_2] = \sum_{i=2}^{12} i \cdot P(X=i | X_1 = X_2) = \frac{2+4+\dots+12}{6} = 7. $$

Here we used that P(X = i|X₁ = X₂) = 1/6 for even i.

3. What is E[X₁|X = 9]?

If X = 9 then there are the only 4 possible values for (X₁, X₂): (3, 6), (4, 5), (5, 4), (6, 3). Thus,

$$ E[X_1 | X=9] = \frac{3 + 4+ 5 + 6}{4} = \frac 9 2 = 4.5. $$

4. What is E[X₁ − X₂|X = k] for k in the range [2, 12]?

Note that X₁ − X₂ ∈ { − 5, 4, …, 5} and that P(X₁ − X₂ = i|X = k) = P(X₁ − X₂ =  − i|X = k) for i ∈ { − 5, 4, …, 5}. Therefore, we have

$$ E[X_1 - X_2 | X=k] = \sum_{i=-5}^5 i \cdot P(X_1-X_2=i | X=k) = 0. $$

num_samples = 10**7

samples = np.random.randint(low=1, high=7, size=(num_samples, 2))

# part (a)
print(f"sampled answer for (a): {samples[samples[:, 0] % 2 == 0, :].sum(axis=1).mean()}")

# part (b)
print(f"\nsampled answer for (b): {samples[samples[:, 0] == samples[:, 1], :].sum(axis=1).mean()}")

# part (c)
print(f"\nsampled answer for (b): {samples[samples.sum(axis=1) == 9][:, 0].mean()}")

# part (d)
print("\nsampled answer for (d)")
for k in range(2, 13):
    cond_samples = samples[samples.sum(axis=1) == k]
    print(f"k= {k}: {(cond_samples[:, 0] - cond_samples[:, 1]).mean()}")

sampled answer for (a): 7.499272007332312

sampled answer for (b): 7.00297383088835

sampled answer for (b): 4.501544227521886

sampled answer for (d)
k= 2: 0.0
k= 3: -0.000497820910275419
k= 4: -0.00014398934478848565
k= 5: 0.0037980995102071674
k= 6: -0.0030300434929995917
k= 7: -0.002319737212486128
k= 8: 0.003350165815196208
k= 9: 0.0030884550437718554
k= 10: -0.0009050957853026369
k= 11: 4.496265401957134e-05
k= 12: 0.0