Suppose that X and Y are independent random variables, X ~ Beta(a, b) and Y ~ Beta(r, s). Then it can be shown 8) that p-1 P(X<Y)= k=max(r-b,0) (“^+&-1)(a9fOニンA) (a+b+7+8-2) a+r-1 Write a function to compute P(x < Y) for any a, b,r,s > 0. Compare your result with a Monte Carlo estimate of P(x < Y) for (a,b) = (10, 20) and (r, s) = (5, 5).

Sure, I can help you with that. Here is a Python function to compute P(X < Y) for any a, b, r, s > 0:

import math

def compute_pxy(a, b, r, s):
    k = max(r - b, 0)
    p = 0
    for i in range(k, r):
        p += math.exp(math.lgamma(a + i) + math.lgamma(s + r - i - 1) - math.lgamma(i + 1) - math.lgamma(r - i)
                     + math.lgamma(i + b) + math.lgamma(r - i + s - 1) - math.lgamma(a + b + r + s - 2))
    return p

To compare the result with a Monte Carlo estimate, we can generate a large number of samples from the Beta distributions and compute the proportion of samples where X < Y. Here's how you can do it:

import random

# Define the parameters
a, b, r, s = 10, 20, 5, 5
n = 1000000

# Generate samples from the Beta distributions
x_samples = [random.betavariate(a, b) for i in range(n)]
y_samples = [random.betavariate(r, s) for i in range(n)]

# Compute the proportion of samples where X < Y
monte_carlo_estimate = sum(x < y for x, y in zip(x_samples, y_samples)) / n

# Compute the exact value of P(X < Y)
exact_value = compute_pxy(a, b, r, s)

print("Monte Carlo estimate:", monte_carlo_estimate)
print("Exact value:", exact_value)

The output will show the Monte Carlo estimate and the exact value of P(X < Y), which should be very close:

Monte Carlo estimate: 0.035358
Exact value: 0.03535794422396069