For ( f(x) = \frac12 |Ax - b|^2 ), derive the closed-form solution for ( x ) that minimizes ( f ). Section 3: Probability & Statistics (20%) Question 7: You have two dice: one fair, one loaded (shows 6 with probability ( 1/3 ), others each ( 2/15 )). You pick a die at random (50% chance each), roll it once, and get a 6. What is the probability it was the loaded die?
Consider ( f(x) = x^3 - 2x + 1 ). Perform one iteration of gradient descent starting at ( x_0 = 1 ) with learning rate ( \eta = 0.1 ). What is ( x_1 )? mbzuai entry exam sample questions
Given vectors ( u = (1, 2, -1) ), ( v = (0, 1, 3) ). Compute the projection of ( u ) onto ( v ). For ( f(x) = \frac12 |Ax - b|^2
Briefly explain how backpropagation computes gradients in a neural network. Why is the chain rule essential? Section 5: Python & Coding Logic (10%) Question 13: What is the output of the following? What is the probability it was the loaded die
def func(x, lst=[]): lst.append(x) return lst print(func(1)) print(func(2)) Write a Python function normalize(X) that takes a 2D numpy array X (samples × features) and returns a zero-mean, unit-variance normalized version (feature-wise). Do not use sklearn .
Given a binary classifier with confusion matrix: TP=90, FN=10, FP=30, TN=70 Compute Precision , Recall , F1-score , and Accuracy .
