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210 |
210 |
211 E[norm(μ, σ²)] = μ |
211 E[norm(μ, σ²)] = μ |
212 |
212 |
213 var[norm(μ, σ²)] = σ² |
213 var[norm(μ, σ²)] = σ² |
214 |
214 |
215 Disjoint distribution of two normal r.v. |
215 Summa of two normal r.v. |
216 ======================================== |
216 ======================== |
217 |
217 |
218 .. math:: |
218 If :math:`Z = X + Y` and X and Y is independent normal r.v. then: |
219 |
219 |
220 norm2(μ₁, μ₂, σ₁², σ₂²) = norm(μ₁, σ₁²)·norm(μ₂, σ₂²) |
220 .. math:: norm(μ_z, σ_z²) = norm(μ_x+μ_y, σ_x²+σ_y²) |
221 |
221 |
222 = 1/(2·π·σ₁·σ₂)·exp(-(x-μ₁)²/σ₁²/2 - (x-μ₂)²/σ₂²/2) |
222 Proof: |
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223 |
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224 .. math:: |
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225 |
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226 norm(μ_z, σ_z²) = ∫_x\ f_X(x)·f_Y(z-x)\ dx |
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227 |
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228 = ∫_x\ 1/sqrt(2·π)/σ_x·exp(-(x-μ_x)²/σ_x²/2)·1/sqrt(2·π)/σ_y·exp(-(z-x-μ_y)²/σ_y²/2)\ dx |
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229 |
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230 = 1/sqrt(2·π·(σ_x² + σ_y²))·exp(-(x-μ_x-μ_y)²/(σ_x²+σ_y²)/2) |
223 |
231 |
224 Linear function of distribution |
232 Linear function of distribution |
225 =============================== |
233 =============================== |
226 |
234 |
227 If :math:`Y = a·X + b` then :math:`f_Y(y) = 1/|a|·f_X((y-b)/a)`. |
235 If :math:`Y = a·X + b` then :math:`f_Y(y) = 1/|a|·f_X((y-b)/a)`. |
285 |
293 |
286 and so: |
294 and so: |
287 |
295 |
288 .. math:: f_Y(y) = (d\ f_Y(t)/dt)(y) = (d\ F_X(h(t))/dt)(y) = F_X(h(y))·(d\ h(t)/dt)(y) |
296 .. math:: f_Y(y) = (d\ f_Y(t)/dt)(y) = (d\ F_X(h(t))/dt)(y) = F_X(h(y))·(d\ h(t)/dt)(y) |
289 |
297 |
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298 Convolution formula |
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299 =================== |
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300 |
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301 If :math:`Z = X + Y` and X and Y is independent r.v. then: |
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302 |
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303 .. math:: f_Z(z) = ∫_x\ f_X(x)·f_Y(z-x)̣·dx |
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304 |
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305 Proof: |
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306 |
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307 Consider :math:`Z` at conditional event :math:`X=x`: |
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308 |
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309 .. math:: f_{Z|X}(z|X=x) = f_{z|X=x}(z|X=x) |
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310 |
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311 Becasue of independence of :math:`X` and :math:`Y`: |
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312 |
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313 .. math:: f_{Z|X}(z|X=x) = f_{X+Y|X=x}(z|X=x) = f_{x+Y}(z) = f_Y(z-x) |
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314 |
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315 Joint PDF of :math:`X` and :math:`Z` is: |
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316 |
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317 .. math:: f_{X,Z}(x,z) = f_X(x)·f_{Z|X}(z|X=x) = f_X(x)·f_Y(z-x) |
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318 |
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319 By integrating by :math:`x` we get: |
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320 |
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321 .. math:: f_Z(z) = ∫_x\ f_{X,Z}(x,z)\ dx = ∫_x\ f_X(x)·f_Y(z-x)\ dx |
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322 |