Evaluate TPR/FPR, sigmoid, and activations

You are taking a 70-minute assessment that includes several multiple-choice questions on ML fundamentals. Work through the items below, showing calculations when they apply.

1) Confusion matrix versus TPR/FPR

Four candidate confusion matrices for binary classification are provided. In each matrix, rows represent Actual and columns represent Predicted.

Option A

• TP=80, FN=20, FP=10, TN=90

Option B

• TP=45, FN=5, FP=45, TN=5

Option C

• TP=50, FN=50, FP=5, TN=95

Option D

• TP=90, FN=10, FP=30, TN=70

Target requirements:

• True Positive Rate (TPR / Recall) = 0.80
• False Positive Rate (FPR) = 0.10

Question: Which option(s) meet both requirements?

2) Sigmoid neuron computation

A single neuron calculates:

$$z = w^\top x + b$$ $$\hat{y} = \sigma(z) = \frac{1}{1+e^{-z}}$$

Given:

• $w = [0.5, -1.0]$
• $x = [2.0, 1.0]$
• $b = 0$

Question: Compute $z$ and $\hat{y}$ to 3 decimal places.

3) Activation functions

Consider these activation functions: sigmoid, tanh, ReLU, Leaky ReLU, softmax.

Questions:

1. Which activation function is most often used in the output layer for multi-class single-label classification?
2. Give two reasons ReLU-like activations are commonly chosen for hidden layers in deep networks.
3. Name one common pitfall of sigmoid/tanh in deep hidden layers and explain why it occurs.

Overview: This question assesses competence in binary classification metrics (confusion matrices, TPR/FPR), basic forward computation in a neural network (sigmoid neuron $z$ and output), and knowledge of activation function selection and pitfalls.