• 👨‍🏫 Yes. As the condition is $\forall x,y,z\in \mathbb{R}$, it's fine

More than 2 r.v.

• joint pdf of multiple r.v. (e.g. $X,Y,Z$) can be described as: $$P((X,Y,Z)\in C)=\int \int {\int }_{(x,y,z)\in C}{f}_{X,Y,Z}(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z$$
• multiple cont. r.v. (e.g. $X,Y,Z$) are independent if, $\forall x,y,z\in \mathbb{R}$: $$f_{X,Y,Z}(x,y,z)=f_X(x)f_Y(y)f_Z(z)$$

Examples

• suppose $X,Y$: independent standard normal dist.
  • find: joint Pdf of $X,Y$
  • from independence: $p_{X,Y}(x,y)=p_X(x)\cdot p_Y(y)$
  • $=\frac{1}{2\pi {\sigma }^2}{e}^{-\frac{x+y}{2}}$
• let $X,Y,Z$ be independent & uniformly distributed over $(0,1)$
  • compute $P(X\ge YZ)$
  • $P(X\ge k)=1-k$
  • $P(YZ=k)=P(Y=y\wedge Z=k/y)=P(Y=y)P(Z=k/y)$ $$\begin{array}{rl}P(X\ge YZ)& =\int \int {\int }_{x\ge yz}{f}_{X,Y,Z}(x,y,z)\mathrm{d}x\mathrm{d}y\mathrm{d}z\\ & ={\int }_0^1{\int }_0^1{\int }_{yz}^{1}1\mathrm{d}x\mathrm{d}y\mathrm{d}z\\ & ={\int }_0^1{\int }_0^{1}1-yz\mathrm{d}y\mathrm{d}z\\ & ={\int }_0^{1}y-y^{2}z/2{|}_0^1\mathrm{d}z\\ & ={\int }_0^{1}1-z/2{|}_0^1\mathrm{d}z\\ & =z-z^2/4{|}_0^1=3/4\end{array}$$

X, Y being continuous & independent

• theorem: on independence, for $x,y\in \mathbb{R}$
  • $f_{X,Y}(x,y)=f_X(x)f_Y(y)$
  • following: $$\begin{array}{rl}{F}_{X,Y}(x)& ={\int }_{-\infty }^{\infty }{F}_X(x-t)f_Y(y)\mathrm{d}t={\int }_{-\infty }^{\infty }{F}_Y(x-t)f_X(t)\mathrm{d}\\ f_{X,Y}(x)& ={\int }_{-\infty }^{\infty }{f}_X(x-t)f_Y(y)\mathrm{d}t={\int }_{-\infty }^{\infty }{f}_Y(t)f_X(x-ty)\mathrm{d}t\end{array}$$
    • ⭐👨‍🏫 important
  • $f_{X+Y}$: "convolution of $f_X,f_Y$"
• proof
• some of 2 important Gamma d.v.
  • $X\sim \Gamma (\alpha ,\lambda )\sim X\sim \Gamma (\alpha ,\lambda )$
  • then $X+Y\sim \Gamma (a+b+\lambda )$
• proof
• ⭐👨‍🏫 sum of independent normal r.v.
  • for $X_i$ where $X$
  • use linearity
• d

X, Y being discrete & independent

• pmf of $X+Y$, each being discrete r.v.: $$\begin{array}{rl}{p}_{X+Y}(t)=P(X+Y=t)& =\sum _{x}P(X=x,Y=t-x)\\ & =p_{X,Y}(x,t-x)\end{array}$$
• similarly, for continuous r.v.: $$\begin{array}{rl}{p}_{X+Y}(t)=P(X+Y=t)& ={\int }_{x}P(X=x,Y=t-x)\mathrm{d}x\\ & =p_{X,Y}(x,t-x)\end{array}$$

Exercises

• $X,Y:indep.w/cmomoncoredist.arcjomoa-$X+t\leq (0,2)$<!---furtherwork-->--$2-x$-remark:if$d_{X+Y}(x,) w/ triangular dist."
  • $X+Y$: dimctop $d$ smooth triangular
• basketball team: plays a 44-game season
  • 26 games: against class A; 18: against class B
  • chance of against class A: $0.4$
  • chance of against class B: $0.7$
  • result of different games: independent
  • $X_A,X_B$: no. of wins against class $A,B$
  • $X_A\sim Bin(26,0.4)$
  • $X_B\sim Bin(18,0.7)$
  • compute $P(X_A+X_B\ge 25)$ or $P(25\le X_A+X_B\le 44)$ (👨‍🏫 either works)
    • 👨‍🏫 use normal approximate
    • $Bin(n,p)\approx N(np,npq)$
    • $X_A\approx N(10.4,6.24)$
    • $X_B\approx N(12.6,3.78)$
    • sum of independent normal: normal dist.
    • $X_A+X_B\approx N(13,10.02)$ $$\begin{array}{rl}P(X_A+X_B\ge 25)& =P(X_A+X_B\ge 24.5)\\ & \approx P(\frac{X_A+X_B-24}{\sqrt{10.02}}\ge \frac{24.5-23}{\sqrt{10.02}})\\ & =P(Z\ge \frac{1.5}{\sqrt{10.02}})\\ & =1-P(Z\le \frac{1.5}{\sqrt{10.02}})\\ & =1-\Phi (\frac{1.5}{\sqrt{10.02}})\approx 0.3178\end{array}$$
  • compute $P(X_A>X_B)$
    • $P(X_A>X_B)=P(X_A\ge X_B+1)=P(X_A-X_B\ge 1)$
    • $-X_B\approx N(-12.6,3.78)$
    • then $X_A-X_B\approx N(10.4-12.6,6.24+3.78)=N(-2.2,10.02)$ $$\begin{array}{rl}P(X_A-X_B\ge 1)& =P(X_A-X_B\ge 0.5)\\ & \approx P(\frac{X_A-X_B+2.2}{\sqrt{10.02}}\ge \frac{0.5+2.2}{\sqrt{10.02}})\\ & =P(Z\ge \frac{2.7}{\sqrt{10.02}})\\ & =1-P(Z\le \frac{2.7}{\sqrt{10.02}})\\ & =1-\Phi (\frac{2.7}{\sqrt{10.02}})\approx 0.1968\end{array}$$
• sum of 2 indep. binomial r.v.
  • $X\sim Bin(n,p);Y\sim Bin(m,p)$, $X,Y$ indep.
  • find pmf of $X+Y$ $$\begin{array}{rl}P(X+Y=k)& =\sum _{i=0}^k\end{array}$$
• s