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Hints for Lab 3: Classical Inference I: Maximum Likelihood Estimation

Hints for Lab 3: Classical Inference I: Maximum Likelihood Estimation#

For the class on Monday, January 22nd

See also

Go back to Lab 3

A. Coin Flips#

Hints for the implementation of Question 1

  • From class we obtained the expression for \(\sigma_p\) in terms of \(N\) and \(K\):

    \[ \sigma_p = \sqrt{\frac{K(N-K)}{N^3}} \]

    We can then rewrite the inequality as \(\hat{p} - 2 \sigma_p > 0.5 \). Plug in the corresponding expressions for \(\hat{p}\) and \(\sigma_p\) such that the left-hand side of the inequality only depends on \(N\) and \(K\).

    Here’s a (not efficient, but easy to understand) way to find the minimal value of \(K\) for the inequality to hold.

    import numpy as np

N = 100
for K in range(1, N+1):
    if ... > 0.5:  # TODO: replace ... with the expression of \hat{p} - 2 \sigma_p but in terms of N and K
        print(K)
        break
Hints for the implementation of Question 2

  • To simulate the number of heads for 100 coin flips, you can simply run

    import numpy as np

n_flips = 100  # number of coin flips
p_heads = 0.5  # heads probability
n_heads = np.random.default_rng().binomial(n_flips, p_heads)
print(n_heads)

    Note that each time you run the code above, n_heads has a different value.

  • The above would be one “simulation”. Repeat this simulation 10,000 time and collect the n_heads from each simulation.

  • You can then use np.percentile to calculate the percentile corresponding to \(2\sigma\) among the 10,000 n_heads values.

B. Estimator Properties#

Hints for “How to verify” 1

  • Here’s a snippet to get you started with the code structure:

    import numpy as np

a = ...  # TODO: set the true value of a
b = ...  # TODO: set the true value of b
N = ...  # TODO: set the number of x's

x = np.random.default_rng().uniform(a, b, size=N)
a_mle = np.min(x)  # this would be \hat{a}
b_mle = np.max(x)  # this would be \hat{b}

# TODO: repeat the above with different values of N and see how a_mle and b_mle change as N increases
Hints for “How to verify” 2

  • Here’s a snippet to get you started with the code structure:

    a = ...  # TODO: set the true value of a
b = ...  # TODO: set the true value of b
N = ...  # TODO: set the number of x's
K = ...  # TODO: set the number of simulations

a_mle_avg = 0
b_mle_avg = 0
for _ in range(K):
    x = np.random.default_rng().uniform(a, b, size=N)
    a_mle_avg += np.min(x)
    b_mle_avg += np.max(x)
a_mle_avg /= K  # this would be $E(\hat{a})$
b_mle_avg /= K  # this would be $E(\hat{b})$

# TODO: repeat the above with different values of K and see how a_mle_avg and b_mle_avg change as K increases
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