Chapter 3 — Simulate in Excel (N = 10)

How to use (applies to all)

1. In Excel set B1 = 10 (sample size N = 10). Put parameters in B2, B3 as shown in each section.
2. Put the generator in D2. If you have 365 with RANDARRAY, it spills to D11. Otherwise, fill down to D11.
3. Compute:
   • Mean: =AVERAGE(D2:D11)
   • Variance (sample): =VAR.S(D2:D11)
4. Quick visual: select D2:D11 → Insert → Column (or Scatter) for a tiny chart.

1) Uniform — continuous on [a, b]

Example: a=2 (B2), b=5 (B3). Values are real numbers between a and b.

Excel generator

365 (spills to 10 rows):

=B2 + (B3-B2)*RANDARRAY(B1,1)

No RANDARRAY (fill D2 → D11):

=B2 + (B3-B2)*RAND()

Theory

• Mean: (a+b)/2
• Variance: (b-a)^2/12

How to recognize it by sight

• Numbers are spread across the entire interval [a, b] with no preference for middle or edges.
• You often see values near both ends, but never outside [a, b].
• A quick histogram looks roughly flat; a simple column chart looks “evenly scattered.”

2) Binomial — X ~ Binomial(n, p)

Example: n=10 (B2), p=0.30 (B3). Values are integers 0…n.

Excel generator

365:

=BINOM.INV(B2,B3,RANDARRAY(B1,1))

No RANDARRAY (fill D2 → D11):

=BINOM.INV(B2,B3,RAND())

Theory

• Mean: n p
• Variance: n p (1-p)

How to recognize it by sight

• All outputs are integers between 0 and n (no decimals).
• Values cluster around n p; extremes like 0 or n are rare unless p is near 0 or 1.
• Shape is symmetric if p≈0.5; skewed toward 0 if p is small, toward n if p is large.

3) Geometric — first success

We’ll use “failures before first success” (support 0,1,2,…). Example: p=0.25 in B2.

Excel generator

365:

=NEGBINOM.INV(1,B2,RANDARRAY(B1,1))

No RANDARRAY (fill D2 → D11):

=NEGBINOM.INV(1,B2,RAND())

Theory (failures counted)

• Mean: (1-p)/p
• Variance: (1-p)/p^2

How to recognize it by sight

• Lots of small numbers (0,1,2 are common) with occasional larger outliers.
• Strong right skew: long “tail” of bigger values happens sometimes.
• All outputs are nonnegative integers.

If you prefer “trials until first success” (support 1,2,…): use the same formula + 1.

=NEGBINOM.INV(1,B2,RANDARRAY(B1,1)) + 1   (365)
=NEGBINOM.INV(1,B2,RAND()) + 1             (fill)

4) Negative Binomial — failures before r-th success

Example: r=3 (B2), p=0.30 (B3).

Excel generator

365:

=NEGBINOM.INV(B2,B3,RANDARRAY(B1,1))

No RANDARRAY (fill D2 → D11):

=NEGBINOM.INV(B2,B3,RAND())

Theory (failures counted)

• Mean: r(1-p)/p
• Variance: r(1-p)/p^2

How to recognize it by sight

• Nonnegative integers with right skew (like geometric), but typically with a larger average and more spread.
• Compared to Poisson, spread (variance) often feels “too big for the mean” (over-dispersion).
• Occasional larger values are not unusual when r and/or (1−p)/p make the mean sizable.

5) Poisson — X ~ Poisson(λ)

Example: λ=4 (B2). Outputs are 0,1,2,… (counts).

Excel generator

365:

=POISSON.INV(B2,RANDARRAY(B1,1))

No RANDARRAY (fill D2 → D11):

=POISSON.INV(B2,RAND())

Theory

• Mean: λ
• Variance: λ (mean ≈ variance)

How to recognize it by sight

• All outputs are nonnegative integers; most values cluster around λ.
• When λ is small (≈1), strong right skew and many zeros; as λ grows (≥5), shape looks more symmetric.
• A quick check: sample variance often near the sample mean (but with N=10 expect noise).

Compare & reflect

1. Compute mean/variance with =AVERAGE(D2:D11) and =VAR.S(D2:D11).
2. Match the “look”:
   • Uniform: evenly spread within [a,b].
   • Binomial: integers 0…n, cluster near np.
   • Geometric: many small values, rare large ones (strong right tail).
   • NegBin: like geometric but larger mean/spread (failures until r successes).
   • Poisson: counts centered near λ; mean≈variance.
3. Write 1–2 sentences: “What clues in my 10 numbers make me think it’s this distribution?”