#### Joint Probability Distribution

p
```"""
Calculate joint probability distribution
https://en.wikipedia.org/wiki/Joint_probability_distribution
"""

def joint_probability_distribution(
x_values: list[int],
y_values: list[int],
x_probabilities: list[float],
y_probabilities: list[float],
) -> dict:
"""
>>> joint_distribution =  joint_probability_distribution(
...     [1, 2], [-2, 5, 8], [0.7, 0.3], [0.3, 0.5, 0.2]
... )
>>> from math import isclose
>>> isclose(joint_distribution.pop((1, 8)), 0.14)
True
>>> joint_distribution
{(1, -2): 0.21, (1, 5): 0.35, (2, -2): 0.09, (2, 5): 0.15, (2, 8): 0.06}
"""
return {
(x, y): x_prob * y_prob
for x, x_prob in zip(x_values, x_probabilities)
for y, y_prob in zip(y_values, y_probabilities)
}

# Function to calculate the expectation (mean)
def expectation(values: list, probabilities: list) -> float:
"""
>>> from math import isclose
>>> isclose(expectation([1, 2], [0.7, 0.3]), 1.3)
True
"""
return sum(x * p for x, p in zip(values, probabilities))

# Function to calculate the variance
def variance(values: list[int], probabilities: list[float]) -> float:
"""
>>> from math import isclose
>>> isclose(variance([1,2],[0.7,0.3]), 0.21)
True
"""
mean = expectation(values, probabilities)
return sum((x - mean) ** 2 * p for x, p in zip(values, probabilities))

# Function to calculate the covariance
def covariance(
x_values: list[int],
y_values: list[int],
x_probabilities: list[float],
y_probabilities: list[float],
) -> float:
"""
>>> covariance([1, 2], [-2, 5, 8], [0.7, 0.3], [0.3, 0.5, 0.2])
-2.7755575615628914e-17
"""
mean_x = expectation(x_values, x_probabilities)
mean_y = expectation(y_values, y_probabilities)
return sum(
(x - mean_x) * (y - mean_y) * px * py
for x, px in zip(x_values, x_probabilities)
for y, py in zip(y_values, y_probabilities)
)

# Function to calculate the standard deviation
def standard_deviation(variance: float) -> float:
"""
>>> standard_deviation(0.21)
0.458257569495584
"""
return variance**0.5

if __name__ == "__main__":
from doctest import testmod

testmod()
# Input values for X and Y
x_vals = input("Enter values of X separated by spaces: ").split()
y_vals = input("Enter values of Y separated by spaces: ").split()

# Convert input values to integers
x_values = [int(x) for x in x_vals]
y_values = [int(y) for y in y_vals]

# Input probabilities for X and Y
x_probs = input("Enter probabilities for X separated by spaces: ").split()
y_probs = input("Enter probabilities for Y separated by spaces: ").split()
assert len(x_values) == len(x_probs)
assert len(y_values) == len(y_probs)

# Convert input probabilities to floats
x_probabilities = [float(p) for p in x_probs]
y_probabilities = [float(p) for p in y_probs]

# Calculate the joint probability distribution
jpd = joint_probability_distribution(
x_values, y_values, x_probabilities, y_probabilities
)

# Print the joint probability distribution
print(
"\n".join(
f"P(X={x}, Y={y}) = {probability}" for (x, y), probability in jpd.items()
)
)
mean_xy = expectation(
[x * y for x in x_values for y in y_values],
[px * py for px in x_probabilities for py in y_probabilities],
)
print(f"x mean: {expectation(x_values, x_probabilities) = }")
print(f"y mean: {expectation(y_values, y_probabilities) = }")
print(f"xy mean: {mean_xy}")
print(f"x: {variance(x_values, x_probabilities) = }")
print(f"y: {variance(y_values, y_probabilities) = }")
print(f"{covariance(x_values, y_values, x_probabilities, y_probabilities) = }")
print(f"x: {standard_deviation(variance(x_values, x_probabilities)) = }")
print(f"y: {standard_deviation(variance(y_values, y_probabilities)) = }")
```  