# MT1A Q2

Subject: Deep Learning

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/#tee/read/10/tee-c10-s2

Canonical: https://deep-learning-study-room.vercel.app/read/dl/tee/section/tee-c10-s2/

## 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.



$$
z=\sum_iw_ix_i+b
$$





$$
a=f(z)
$$



**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.



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



**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.**



$$
x_1w_1=2(0.4)=0.8
$$



**Step 2: multiply input 2 by weight 2.**



$$
x_2w_2=1(-0.2)=-0.2
$$



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

**Step 3: multiply input 3 by weight 3.**



$$
x_3w_3=3(0.5)=1.5
$$



**Step 4: add the three contributions.**



$$
s=0.8+(-0.2)+1.5=0.6+1.5=2.1
$$



**Step 5: add the bias once.**



$$
z=s+b=2.1+0.1=2.2
$$



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

**Step 6: negate z for the exponential.**



$$
-z=-2.2
$$





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



**Step 7: calculate the complete denominator.**



$$
1+e^{-z}=1+0.110803158=1.110803158
$$



**Step 8: divide one by that denominator.**



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



**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.