MATH 2421: Lecture 19

Date: 2024-11-11 11:54:21

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Joint PDF of Functions of RV

Generalization to multiple variables
- 👨‍🏫 for exams: you are expected to do max. $2$
- Jacobian determinant expands, to $n \times n$
Example
- let $X_{1}, X_{2}$ : jointly continuous r.v. w/ pdf $f_{X_{1}, X_{2}}$
  - let $Y_{1} = X_{1} + X_{2}, Y_{2} = X_{1} - X_{2}$
  - find joint density function of $Y_{1}, Y_{2}$ in terms of $f_{X_{1}, X_{2}}$
  - d $f_{Y_{1}, Y_{2}} (y_{1}, y_{2}) = f_{X_{1}, X_{2}} (\frac{1}{2} (y_{1} + y_{2}), \frac{1}{2} (y_{1} - y_{2})) \frac{\partial ( y _{1} , y _{2} )}{\partial ( x _{1} , x _{2} )}^{- 1} = \frac{1}{2} f_{X_{1}, X_{2}} (\frac{1}{2} (y_{1} + y_{2}), \frac{1}{2} (y_{1} - y_{2}))$
- suppose $X, Y$ : independent standard normal variable
  - show: $X + Y, X - Y$ are independent normal variables
  - let $X_{1} = X; X_{2} = Y$
  - and $Y_{1} = X + Y; Y_{2} = X - Y$
  - as they are independent: $f_{X_{1}, X_{2}} (x_{1}, x_{2}) = f_{X_{1}} (x_{1}) f_{X_{2}} (x_{2}) = \frac{1}{2 π} e^{- \frac{x _{1}^{2} + x _{2}^{2}}{2}}$
  - by previous example: joint pdf of $Y_{1}, Y_{2}$ is $f_{Y_{1}, Y_{2}} (y_{1}, y_{2}) = \frac{1}{2} f_{X_{1}, X_{2}} (\frac{1}{2} (y_{1} + y_{2}), \frac{1}{2} (y_{1} - y_{2})) = \frac{1}{2} \frac{1}{2 π} e^{- \frac{1}{2} (\frac{1}{4} (y_{1} + y_{2})^{2} + \frac{1}{4} (y_{1} - y_{2})^{2})} = \frac{1}{4 π} e^{- \frac{1}{2} (\frac{1}{2} y_{1}^{2} + \frac{1}{2} y_{2}^{2})} = \frac{1}{4 π} e^{- \frac{1}{4} (y_{1}^{2} y_{2}^{2})} = \frac{1}{4 π} e^{- \frac{1}{4} y_{1}^{2}} \frac{1}{4 π} e^{- \frac{1}{4} y_{2}^{2}}$
  - as con
  - $Y_{1} \sim N (0, 2)$
- let $X, Y$ be independent standard normal
  - then $R = X^{2} + Y^{2}, Θ = arctan (\frac{Y}{X})$
  - $f_{X, Y} (x, y) = \frac{1}{2 π} e^{- \frac{x ^{2} + y ^{2}}{2}}$ ${r = x^{2} + y^{2} Θ = arctan (\frac{y}{x}) {x = r cos Θ y = r sin Θ$
  - pdf of $R, Θ$
  - domain: $r > 0 \cap Θ \in [0, 2 π)$ $\frac{\partial ( x , y )}{\partial ( r , Θ )} = \frac{\partial x}{\partial r} \frac{\partial y}{\partial r} \frac{\partial x}{\partial Θ} \frac{\partial y}{\partial Θ} = cos Θ sin Θ - r sin Θ r cos Θ = r cos^{2} - (- r sin^{2}) = r \geq 0$
  - thus, pdf of $R, Θ$ : $f_{R, Θ} (r, Θ) = f_{X, Y} (x, y) = \frac{1}{2 π} e^{- \frac{r ^{2}}{2}} r$
  - for joint pdf $f_{R, Θ} (r, θ)$ is factorization, $R, Θ$ are independent
    - $Θ$ : uniform distribution
      - 👨‍🏫 rotation ignorance property of normal vectors
  - $R$ : Rayleigh distribution
- if $X, Y$ : independent Gamma r.v. w/ parameters $(α, γ), (β, γ)$
  - compute joint density of $U = X + Y; V = X / (X + Y)$ ${u = x + y v = \frac{x}{x + y} ⟹ {x = uv y = u - uv = u (1 - v)$
- Finally, joint pdf of $X, Y$ : $f_{X, Y (x . y)} = f_{X} (x) f_{Y} (y) = A ^{α + β}$ $\frac{λ e ^{- λ u} ( λ u ) ^{α + β + 1}}{Γ ( α + β )}$
- try harder later

Jointly Distributed R.V w/ n>2

Jointly distributed r.v.
- $F_{X, Y, Z} (x, y, z) := P (X \leq x, Y \leq y, Z \leq z)$
- marginal distribution, namely:
  - $F_{X, Y} (x, y) := lim_{z \to \infty} F_{X, Y, Z} (x, y, z)$
- similar for density function

Expectation of Sum of Random Variables

Expectation of sum of random variables
- theorem: $a \leq X \leq b ⟹ a \leq E (X) \leq b$
- proof:
- if $X, Y$ : jointly discrete w/ joint pmf $p_{X, Y}$
  - $E [g (X, Y)] = \sum_{y} \sum_{x} g (x, y) p_{X, Y} (x, y)$
- if $X, Y$ : jointly continuous w/ joint pdf $p_{X, Y}$
  - $E [g (X, Y)] = \int_{y} \int_{x} g (x, y) p_{X, Y} (x, y) d x d y$
- remarks
  1. if $g (x, y) \geq 0$ whenever $p_{X, Y} > 0$ , then $E [g (X, Y)] \geq 0$
  2. $E [g (X, Y) + h (X, Y)] = E [g (X, Y)] + E [h (X, Y)]$
  3. $E [g (X) + h (Y)] = E [g (X)] + E [h (Y)]$
  4. Monotone property, if $X \leq Y$ , then $E (X) \leq E (Y)$
- important special case
  - mean of sum: sum of means
  - $E (X + Y) = E (X) + E (Y)$
    - leads to: linearity of expectation regardless of independency
  - to compute the sum: marginal pdf / pmf is enough
- d
Example
- accident: at point $X$ w/ uniformed distributed on a road of length $L$
  - ambulance: at location $Y$ , uniformly distributed on the same road
  - find: expected distance between the ambulance and point of the accident
  - $X \sim U (0, L); Y \sim Y (0, L)$
  - joint pdf: multiplication! $f_{X, Y} (x, y) = {\frac{1}{L ^{2}} 0 if x \in (0, L), y \in (0, L) otherwise$ $E [∣ X - Y ∣] = \int\int ∣ x - y ∣ f_{X, Y} (x, y) d x d y = \int_{0}^{L} \int_{0}^{L} ∣ x - y ∣ \frac{1}{L ^{2}} d x d y = 2 \int_{0}^{L} \int_{y}^{L} (x - y) \frac{1}{L ^{2}} d x d y = 2 \int_{0}^{L} (\frac{x ^{2}}{2 L ^{2}} - \frac{x y}{L ^{2}}) ∣_{y}^{L} d y = 2 \int_{0}^{L} \frac{1}{2} - \frac{y}{L} + \frac{y ^{2}}{2 L ^{2}} d y = 2 (\frac{y}{2} - \frac{y ^{2}}{2 L} + \frac{y ^{3}}{6 L ^{2}}) ∣_{0}^{L} = 2 (\frac{L}{2} - \frac{L}{2} + \frac{L}{6}) = \frac{L}{3}$
- sample mean: let $X_{1}, \dots, X_{n}$ : independent & identically distributed r.v.
  - w/ distribution function $F$ and expected value $μ$
  - such sequence of r.v.: constitute a sample from distribution $F$
  - sample mean $\overset{ˉ}{X}$ : defined as $\overset{ˉ}{X} = \frac{1}{n} k = 1 \sum n X_{k}$
  - $E (\overset{ˉ}{X}) = E [\frac{1}{n} \sum_{k = 1}^{n} X_{k}] = \frac{1}{n} \sum_{k = 1}^{n} E [X_{k}] = \frac{1}{n} \sum_{k = 1}^{n} μ = μ$
- Boole's inequality (skipped)
- mean of hypergeometric
  - $n$ balls: selected from $N$ balls of which $m$ are white
    - find expected no. of white balls selected
  - $X$ : no. of white balls selected
  - use indicator random variable
    - $E (X) = \sum_{k} E (X_{k}) = \sum_{k} P (x_{k} = 1)$
  - $E (X_{i}) = \frac{m}{N}$
  - $E (X) = n \cdot \frac{m}{N}$

MATH 2421: Probability