Evaluate TPR/FPR, sigmoid, and activations

You have a 70-minute assessment with several ML-fundamentals multiple-choice questions. Answer the following (show calculations where applicable).

1) Confusion matrix vs. TPR/FPR

You are given four candidate confusion matrices (binary classification). For each matrix, rows are Actual and columns are 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) satisfy both requirements?

2) Sigmoid neuron calculation

A single neuron computes:

$z = w^\top x + b$

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

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 the following activation functions: sigmoid, tanh, ReLU, Leaky ReLU, softmax.

Questions:

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