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ProbTheory: Make the sum of gaussians interactive #298

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ProbTheory: Make the sum of gaussians interactive
fonsp Aug 20, 2026
6d3cc33
Update Probability Theory Review.jl
fonsp Aug 20, 2026
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113 changes: 68 additions & 45 deletions lectures/Probability Theory Review.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -1107,51 +1107,16 @@ md"""
Set the parameters for the distributions of ``X`` and ``Y``:
"""

# ╔═╡ 27ec154a-a4c3-4d71-b2a0-45f2b456a8e4
μx = 2.0; σx = 1.0;

# ╔═╡ de4dbfc9-9340-4ae2-b323-49abfd77f198
μy = 2.0; σy = 0.5;

# ╔═╡ 1cb8b2c4-e1ae-4973-ba53-fc6c7fe1f37a
md"""
Compute the parameters for the distribution of ``Z = X + Y``:
"""

# ╔═╡ 91a91472-ee6d-416b-b18e-acbedc03a7fe
μz = μx + μy

# ╔═╡ 6485575d-c5a5-4891-8210-f50d6f75476f
σz = sqrt(σx^2 + σy^2)

# ╔═╡ 0abaed25-decc-4dcd-aa04-b68ec0d5c73e


# ╔═╡ 218d3b6e-50b6-4b98-a00c-a19dd33d2c03
md"""
Let's plot the distributions for ``X``, ``Y``, and ``Z``
Let's plot the distributions for ``X``, ``Y``, and ``Z``.
"""

# ╔═╡ e836f877-5ed6-4865-ba3a-1ca5a86b2349
begin
x = Normal(μx, σx)
y = Normal(μy, σy)
z = Normal(μz, σz)
end;

# ╔═╡ c0ea3253-a06b-426c-91a3-a6dd33e42779
let

# Calculate the x-range for plotting
range_min = min(μx-2*σx, μy-2*σy, μz-2*σz)
range_max = max(μx+2*σx, μy+2*σy, μz+2*σz)
range_grid = range(range_min, stop=range_max, length=100)

plot(range_grid, t -> pdf(x,t), label=L"p_x", fill=(0, 0.1))
plot!(range_grid, t -> pdf(y,t), label=L"p_y", fill=(0, 0.1))
plot!(range_grid, t -> pdf(z,t), label=L"p_z", fill=(0, 0.1))
end

# ╔═╡ 3e1f4f46-d294-11ef-29b8-69e546763781
md"""
## PDF for the Product of Two Variables
Expand Down Expand Up @@ -1494,6 +1459,67 @@ Many people have trouble distinguishing ``p(A|B)`` from ``p(B|A)`` in their head
""")
end

# ╔═╡ 577c6e08-9b74-43a4-9e27-9c477980406b
function NormalBind(bond_mu, bond_sigma; style_css::String="")
PlutoUI.Experimental.transformed_value(
function((x,y))
Normal(x,y)
end,
PlutoUI.combine() do Child
@htl """
<code style='text-wrap: nowrap; background: none; $(style_css)'>Normal(μ=$(Child(bond_mu)), σ=$(Child(bond_sigma)))</code>
"""
end
)
end

# ╔═╡ d1ab6fc4-e92c-4e77-8c4a-f5f33cb0b52c
@htl """
<style>
.psg {
display: block;
padding: 1rem;
border-radius: 1rem;
border: 3px solid gray;
color: black;
}
</style>

<div style='display: flex; gap: 1rem;'>
<div class=psg style='background: #eef'>
<h3>Parameters for X</h3>
<p>$(@bind x NormalBind(Scrubbable(2.0), Scrubbable(1.0)))</p>
</div>

<div class=psg style='background: #fee'>
<h3>Parameters for Y</h3>
<p>$(@bind y NormalBind(Scrubbable(2.0), Scrubbable(0.5)))</p>
</div>
</div>
"""

# ╔═╡ 34f1f3cb-9346-4bd3-ae9f-a25b4d5de781
z = Normal(
x.μ + y.μ, # mean
√(x.σ^2 + y.σ^2) # stdev
)

# ╔═╡ c0ea3253-a06b-426c-91a3-a6dd33e42779
let

# Calculate the x-range for plotting
# range_min = min(μx-2*σx, μy-2*σy, μz-2*σz)
# range_max = max(μx+2*σx, μy+2*σy, μz+2*σz)
range_min = 0
range_max = 6
range_grid = range(range_min, stop=range_max, length=100)

plot(ylim=(0,1))
plot!(range_grid, t -> pdf(x,t), label=L"X", lw=2, fill=(0, 0.1))
plot!(range_grid, t -> pdf(y,t), label=L"Y", lw=2, fill=(0, 0.1))
plot!(range_grid, t -> pdf(z,t), label=L"Z", lw=2, fill=(0, 0.1), linestyle=:dash)
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
Expand All @@ -1516,7 +1542,7 @@ Plots = "~1.41.6"
PLUTO_MANIFEST_TOML_CONTENTS = """
# This file is machine-generated - editing it directly is not advised

julia_version = "1.12.6"
julia_version = "1.12.7"
manifest_format = "2.0"
project_hash = "0fd20a2ca9903be4db968c9668bf0e2c12e12de3"

Expand Down Expand Up @@ -1621,7 +1647,7 @@ version = "1.0.2"
[[deps.CompilerSupportLibraries_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae"
version = "1.3.0+1"
version = "1.3.1+2"

[[deps.ConcurrentUtilities]]
deps = ["Serialization", "Sockets"]
Expand Down Expand Up @@ -2129,7 +2155,7 @@ version = "1.6.1"
[[deps.OpenSSL_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
version = "3.5.4+0"
version = "3.5.6+0"

[[deps.OpenSpecFun_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl"]
Expand Down Expand Up @@ -2908,14 +2934,10 @@ version = "1.13.0+0"
# ╟─3e1eeb14-d294-11ef-1702-f5d2cf6fe60a
# ╟─3e1f225a-d294-11ef-04c6-f3ca018ab286
# ╟─98fa17a6-7c8b-46e4-b32d-52db183d88f8
# ╠═27ec154a-a4c3-4d71-b2a0-45f2b456a8e4
# ╠═de4dbfc9-9340-4ae2-b323-49abfd77f198
# ╟─d1ab6fc4-e92c-4e77-8c4a-f5f33cb0b52c
# ╟─1cb8b2c4-e1ae-4973-ba53-fc6c7fe1f37a
# ╠═91a91472-ee6d-416b-b18e-acbedc03a7fe
# ╠═6485575d-c5a5-4891-8210-f50d6f75476f
# ╟─0abaed25-decc-4dcd-aa04-b68ec0d5c73e
# ╠═34f1f3cb-9346-4bd3-ae9f-a25b4d5de781
# ╟─218d3b6e-50b6-4b98-a00c-a19dd33d2c03
# ╠═e836f877-5ed6-4865-ba3a-1ca5a86b2349
# ╟─c0ea3253-a06b-426c-91a3-a6dd33e42779
# ╟─3e1f4f46-d294-11ef-29b8-69e546763781
# ╟─3e1f68fa-d294-11ef-31b2-e7670da8c08c
Expand Down Expand Up @@ -2951,5 +2973,6 @@ version = "1.13.0+0"
# ╠═42b47af6-b850-4987-a2d7-805a2cb64e43
# ╠═a66ab9df-897c-42e5-8b0f-c520ceaffa23
# ╠═4a81342c-17c7-4eb9-933b-edb98df7b9c4
# ╟─577c6e08-9b74-43a4-9e27-9c477980406b
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
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