Deep Learning · MT2 · section

MT1A Q2

Prerequisite repair and teacher backprop examples

Question

Question (3 marks). A sigmoid neuron has inputs [2,1,3], weights [0.4,−0.2,0.5] and bias 0.1. Compute its output.

Answer — step by step

Given

Inputs x₁ = 2,x₂ = 1,x₃ = 3.

Matching weights w₁ = 0.4,w₂ = −0.2,w₃ = 0.5.

Bias b = 0.1; activation = sigmoid.

To find

Calculate net input z and activated output a.

Formula — which one and why

Weighted sum and activation. Each input is multiplied by its connection weight. Add these contributions, add the bias once, then apply the stated activation.

xᵢ = input i; wᵢ = its weight; b = bias; z = sum before activation; a = output after activation; f = the specified activation function.

z=\sum_iw_ix_i+bLaTeX: z=\sum_iw_ix_i+b
a=f(z)LaTeX: a=f(z)

Yaad rakhne ka cue: Input × weight, sabka sum, bias once, phir activation.

Sigmoid output. The question specifies sigmoid, so turn the complete weighted sum into an output between 0 and 1. Negate z, evaluate the exponential, add 1, then take the reciprocal.

z = sum before activation; a = sigmoid output; e ≈ 2.71828 is the exponential base.

a=\frac1{1+e^{-z}}LaTeX: a=\frac1{1+e^{-z}}

Yaad rakhne ka cue: Minus-z ki e power, plus one, phir one divided by whole denominator.

Solution — step by step

Step 1: multiply input 1 by weight 1.

x_1w_1=2(0.4)=0.8LaTeX: x_1w_1=2(0.4)=0.8

Step 2: multiply input 2 by weight 2.

x_2w_2=1(-0.2)=-0.2LaTeX: x_2w_2=1(-0.2)=-0.2

The minus sign belongs to the weight, so the contribution stays negative.

Step 3: multiply input 3 by weight 3.

x_3w_3=3(0.5)=1.5LaTeX: x_3w_3=3(0.5)=1.5

Step 4: add the three contributions.

s=0.8+(-0.2)+1.5=0.6+1.5=2.1LaTeX: s=0.8+(-0.2)+1.5=0.6+1.5=2.1

Step 5: add the bias once.

z=s+b=2.1+0.1=2.2LaTeX: z=s+b=2.1+0.1=2.2

This is the preactivation. It is not yet the requested sigmoid output.

Step 6: negate z for the exponential.

-z=-2.2LaTeX: -z=-2.2
e^{-z}=e^{-2.2}\approx0.110803158LaTeX: e^{-z}=e^{-2.2}\approx0.110803158

Step 7: calculate the complete denominator.

1+e^{-z}=1+0.110803158=1.110803158LaTeX: 1+e^{-z}=1+0.110803158=1.110803158

Step 8: divide one by that denominator.

a=\frac1{1+e^{-z}}=\frac1{1.110803158}\approx0.900249511LaTeX: a=\frac1{1+e^{-z}}=\frac1{1.110803158}\approx0.900249511

Final answer: sigmoid output ≈ 0.900249511. It lies between 0 and 1. The raw sum was 2.2, whereas the activated output is approximately 0.90025.