Practice Questions — Mathematics for Machine Learning

import numpy as np
a = np.array([1, 2, 3])
b = np.array([4, 5, 6])
# Manual: 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32
print(f"Dot product: {np.dot(a, b)}")

import numpy as np
I = np.eye(3)
print(I)

import numpy as np
A = np.array([[1, 2], [3, 4]])
B = np.array([[5, 6], [7, 8]])
C = A @ B
print(f"A @ B:\n{C}")
print(f"\nElement [0][1] = 1*6 + 2*8 = {C[0, 1]}")

import numpy as np
arr = np.array([10, 20, 30, 40, 50, 1000])
print(f"Mean: {np.mean(arr):.1f}")
print(f"Median: {np.median(arr):.1f}")
print(f"Std: {np.std(arr):.1f}")
print("Mean is 191.7, median is 35.0")
print("The outlier 1000 drags the mean up but does not affect the median")

x = 5.0
learning_rate = 0.1
gradient = 2 * x  # derivative of x^2
x_new = x - learning_rate * gradient
print(f"Old x: {x}")
print(f"Gradient at x=5: {gradient}")
print(f"New x: {x_new}")
print(f"f(old x) = {x**2}, f(new x) = {x_new**2}")

import numpy as np

X = np.array([1, 2, 3, 4, 5])
Y = np.array([2, 4, 5, 4, 5])

# Manual calculation
mean_x, mean_y = np.mean(X), np.mean(Y)
cov_xy = np.mean((X - mean_x) * (Y - mean_y))
std_x, std_y = np.std(X), np.std(Y)
corr_manual = cov_xy / (std_x * std_y)

# NumPy verification
corr_numpy = np.corrcoef(X, Y)[0, 1]

print(f"Manual correlation: {corr_manual:.4f}")
print(f"NumPy correlation: {corr_numpy:.4f}")

x = 0.0
lr = 0.2

for i in range(20):
    grad = 2 * (x - 5)
    x = x - lr * grad

print(f"Final x: {x:.6f} (expected: 5.0)")
print(f"Final f(x): {(x-5)**2 + 3:.6f} (expected: 3.0)")

import numpy as np
A = np.array([[2, 1], [5, 3]])
A_inv = np.linalg.inv(A)
print(f"A:\n{A}")
print(f"\nA inverse:\n{A_inv}")
print(f"\nA @ A_inv:\n{np.round(A @ A_inv)}")
print(f"Is identity? {np.allclose(A @ A_inv, np.eye(2))}")

p_six = 1 / 6
p_not_six = 1 - p_six
print(f"P(rolling 6): {p_six:.4f} ({p_six*100:.2f}%)")
print(f"P(not rolling 6): {p_not_six:.4f} ({p_not_six*100:.2f}%)")

p_disease = 0.02
p_healthy = 0.98
p_pos_disease = 0.95
p_pos_healthy = 0.10

p_pos = p_pos_disease * p_disease + p_pos_healthy * p_healthy
p_disease_pos = (p_pos_disease * p_disease) / p_pos

print(f"P(disease | positive test): {p_disease_pos:.4f}")
print(f"That's only {p_disease_pos*100:.1f}%!")

import numpy as np

hours = np.array([2, 4, 6, 8, 3, 7, 5])
attendance = np.array([60, 70, 85, 95, 65, 90, 80])
marks = np.array([50, 65, 80, 92, 55, 85, 72])

corr = np.corrcoef([hours, attendance, marks])
labels = ['Hours', 'Attend', 'Marks']

print('Correlation Matrix:')
print(f"{'':>10}", end='')
for l in labels: print(f"{l:>10}", end='')
print()
for i, l in enumerate(labels):
    print(f"{l:>10}", end='')
    for j in range(3):
        print(f"{corr[i][j]:>10.4f}", end='')
    print()

import numpy as np
a = np.array([3, 4])
b = np.array([1, 2])
print(f"a + b = {a + b}")
print(f"a - b = {a - b}")
print(f"a * 2 = {a * 2}")
print(f"dot product = {np.dot(a, b)}")

import numpy as np
v = np.array([3, 4])
mag = np.linalg.norm(v)
print(f"Vector: {v}")
print(f"Magnitude: {mag}")
print(f"Manual: sqrt(3^2 + 4^2) = sqrt(9+16) = sqrt(25) = {np.sqrt(9+16)}")

import numpy as np

w = np.array([5.0, 5.0])
lr = 0.1

for i in range(30):
    grad = 2 * w
    w = w - lr * grad
    if i % 5 == 0:
        print(f"Step {i:2d}: w = [{w[0]:.4f}, {w[1]:.4f}], f(w) = {np.sum(w**2):.6f}")

print(f"\nFinal: w = [{w[0]:.6f}, {w[1]:.6f}]")
print(f"Expected: [0, 0]")

import numpy as np

def normal_equation(X, y):
    return np.linalg.inv(X.T @ X) @ X.T @ y

# Simple data: y = 2*x + 1
X = np.array([[1, 1], [1, 2], [1, 3], [1, 4], [1, 5]])  # Bias column + feature
y = np.array([3, 5, 7, 9, 11])  # y = 2x + 1

w = normal_equation(X, y)
print(f"Weights: bias={w[0]:.2f}, slope={w[1]:.2f}")
print(f"Equation: y = {w[1]:.2f}x + {w[0]:.2f}")

import numpy as np
np.random.seed(42)

flips = np.random.choice(['H', 'T'], size=10000)
heads = np.sum(flips == 'H')
p_heads = heads / len(flips)

print(f"Total flips: {len(flips)}")
print(f"Heads: {heads}")
print(f"P(Heads): {p_heads:.4f} (expected: 0.5000)")