Normal Random Variables (Adding, Subtracting, Multiplying, Dividing)

Info sheet · Statistics for Psychology & Neuroscience

Author

Andrew Bell

Published

August 13, 2026

Info sheet 0.2 (draft) · Prerequisites: the normal distribution; mean and variance · Give feedback ↗

✎ Editing notes — to do / to check

Working notes for the author — not shown to students once collapsed; remove before publishing.

What you’ll get from this sheet

What happens when you combine normal random variables — and when the result stays a bell curve. By the end you should be able to:

Give the mean and variance of \(aX+b\), and of \(X \pm Y\).
Say why sums and differences of independent normals are still normal, but products and quotients are not.

Scale/shift a normal and it stays normal: \(aX+b \sim N(a\mu+b,\; a^2\sigma^2)\). Add or subtract two independent normals and it stays normal, with the variances adding either way: \(X\pm Y \sim N(\mu_X\pm\mu_Y,\; \sigma_X^2+\sigma_Y^2)\). Multiply or divide them and the result is not normal.

The rules

A normal random variable \(X \sim N(\mu,\sigma^2)\) is defined by its mean and variance. Combine normals and some operations preserve the bell shape while others don’t:

Linear transform. \(aX+b \sim N(a\mu+b,\; a^2\sigma^2)\). Shifting by \(b\) moves the mean; scaling by \(a\) multiplies the SD by \(|a|\) (so the variance by \(a^2\)). Still normal.
Sum / difference (independent \(X,Y\)). \(X\pm Y \sim N(\mu_X\pm\mu_Y,\; \sigma_X^2+\sigma_Y^2)\). The means add or subtract as you’d expect — but the variances add in both cases. Subtracting doesn’t cancel uncertainty; it compounds it. Still normal.
Product / quotient. \(X\times Y\) and \(X/Y\) are not normal. A product of normals is skewed; a quotient is heavy‑tailed (near‑Cauchy, because the denominator can land near zero). No neat mean/variance shortcut applies.

See which operations stay normal

Two independent normals, \(X \sim N(3,1)\) and \(Y \sim N(1,1)\). Pick an operation and watch the resulting distribution — the green curve is the normal the rules predict (drawn only when the result really is normal):

md`**Operation**`

viewof op = html`<select style="font-size:15px;padding:4px 8px">
  <option value="add">X + Y</option>
  <option value="sub" selected>X − Y</option>
  <option value="lin">2X + 3</option>
  <option value="mul">X × Y</option>
  <option value="div">X ÷ Y</option>
</select>`

nrvWidget = {
  let s=8;
  const rnd=()=>{s|=0;s=s+0x6D2B79F5|0;let t=Math.imul(s^s>>>15,1|s);t=t+Math.imul(t^t>>>7,61|t)^t;return((t^t>>>14)>>>0)/4294967296;};
  const nrm=(m,sd)=>{const u=1-rnd(),v=rnd();return m+sd*Math.sqrt(-2*Math.log(u))*Math.cos(2*Math.PI*v);};
  const N=8000, out=[];
  for(let i=0;i<N;i++){const X=nrm(3,1), Y=nrm(1,1);
    out.push(op==="add"?X+Y: op==="sub"?X-Y: op==="lin"?2*X+3: op==="mul"?X*Y: X/Y);}
  // predicted normal params where valid
  const pred = op==="add"?{m:4,v:2}: op==="sub"?{m:2,v:2}: op==="lin"?{m:9,v:4}: null;
  const isNormal = pred!==null;
  // clip range for display (div/mul have heavy tails)
  const sorted=[...out].sort((a,b)=>a-b);
  const lo=sorted[Math.floor(N*0.01)], hi=sorted[Math.floor(N*0.99)];
  const bins=44, bw=(hi-lo)/bins, c=Array(bins).fill(0);
  out.forEach(v=>{let b=Math.floor((v-lo)/bw); if(b>=0&&b<bins)c[b]++;});
  const W=470,H=280,pad=30, plotH=H-2*pad, mx=Math.max(...c);
  const X=v=>pad+(v-lo)/(hi-lo)*(W-2*pad), Y=k=>H-pad-k/mx*plotH;
  let bars=c.map((k,i)=>`<rect x="${X(lo+i*bw)}" y="${Y(k)}" width="${(W-2*pad)/bins-1}" height="${H-pad-Y(k)}" fill="${isNormal?'#1F4E79':'#C0501E'}" fill-opacity="0.4"/>`).join("");
  let overlay="";
  if(isNormal){const sd=Math.sqrt(pred.v); const pdf=x=>Math.exp(-((x-pred.m)**2)/(2*pred.v))/(sd*Math.sqrt(2*Math.PI));
    const scale=mx/ (Math.max(...c.map((k,i)=>pdf(lo+(i+0.5)*bw)))||1);
    let p="M "; for(let x=lo;x<=hi;x+=(hi-lo)/160) p+=`${X(x)} ${Y(pdf(x)*scale)} `; overlay=`<path d="${p}" fill="none" stroke="#1E7A34" stroke-width="2.5"/>`;}
  const meanObs=out.reduce((a,b)=>a+b,0)/N;
  const caps={add:"Sum of independent normals → <b>N(4, 2)</b>: means add (3+1), variances add (1+1). Still normal.",
    sub:"Difference of independent normals → <b>N(2, 2)</b>: means subtract (3−1), but variances still <b>add</b> (1+1) — subtracting doesn't cancel uncertainty!",
    lin:"Linear transform → <b>N(9, 4)</b>: mean 2·3+3, variance 2²·1. Still normal.",
    mul:"Product X×Y is <b>not normal</b> — it's skewed. Mean is μ<sub>X</sub>μ<sub>Y</sub>=3, but no tidy normal describes it.",
    div:"Quotient X÷Y is <b>not normal</b> — heavy‑tailed (near‑Cauchy), because Y can land close to zero. Mean and variance may not even exist."}[op];
  const fig=document.createElement("figure"); fig.style.margin="0";
  fig.innerHTML=`<svg viewBox="0 0 ${W} ${H}" width="${Math.min(W,540)}" style="background:#f8fafc;border:1px solid #e2e8f0;border-radius:8px"><line x1="${pad}" y1="${H-pad}" x2="${W-pad}" y2="${H-pad}" stroke="#94a3b8"/>${bars}${overlay}</svg>`+
    `<figcaption style="margin-top:.5rem">${caps} <span style="color:#94a3b8">(simulated mean ≈ ${meanObs.toFixed(2)}).</span> ${isNormal?'<span style="color:#1E7A34">Green curve = the predicted normal — it fits.</span>':'<span style="color:#c0392b">No normal curve drawn — the result is not normal.</span>'}</figcaption>`;
  return fig;
}

The sum, difference, and linear transform all produce a clean bell curve that the green predicted‑normal traces exactly — and note the difference case: \(X - Y\) has variance \(1 + 1 = 2\), not zero. Uncertainties compound whether you add or subtract, which is why a difference score is noisier than either measurement alone. Switch to the product or quotient and the histogram goes lopsided or grows fat tails, and no normal curve fits — those operations leave the normal family entirely.

See it in code

X = 3 + randn(1e5,1);  Y = 1 + randn(1e5,1);
[mean(X-Y), var(X-Y)]     % ~ (2, 2)
[mean(2*X+3), var(2*X+3)] % ~ (9, 4)
histogram(X./Y, 200)      % quotient: heavy-tailed, not normal

The R and Python tabs run live; MATLAB is a static reference.

Check your understanding

Pre‑ and post‑test scores are each \(N(50, 100)\) and independent. What’s the distribution of the change score (post − pre)?

\(N(0,\; 200)\). The means subtract (50 − 50 = 0), but the variances add (100 + 100 = 200), so the change score has SD \(\sqrt{200}\approx 14.1\) — larger than either test’s SD of 10. That’s the key, counter‑intuitive point: taking a difference doesn’t cancel noise, it accumulates it, which is exactly why difference/change scores are noisier and why paired designs analyse them carefully.

“Variances add” only holds for independent variables — if \(X\) and \(Y\) are correlated, the variance of \(X\pm Y\) picks up a \(\pm 2\,\text{Cov}(X,Y)\) term (which is precisely how a paired t‑test gains power from positively correlated pre/post scores). And don’t assume every function of normals is normal: products and quotients aren’t, so you can’t slap a mean‑and‑SD normal onto a ratio of measurements.

Where this shows up next

These rules underlie standard errors, the CLT, and why paired designs beat independent ones. See Chapter (Foundations) and the Normal Distribution & CLT sheet.

What you’ll get from this sheet

The rules

See which operations stay normal

See it in code

Where this shows up next

Related topics