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# Lecture 24 ## ECE 486 ## Spring 2024 ### Generating Random Variables * **Problem**: How can we generate a random variable $Y=g(U)$, where $U \sim$ Uniform(0,1), to follow a prescribed CDF $F_Y$? * **Solution**: $Y = F_Y^{-1}(U)$ ### Proof $F_Y(y) = P(Y \le y) = P(F_Y^{-1}(U) \le y) = P(...

# Lecture 24 ## ECE 486 ## Spring 2024 ### Generating Random Variables * **Problem**: How can we generate a random variable $Y=g(U)$, where $U \sim$ Uniform(0,1), to follow a prescribed CDF $F_Y$? * **Solution**: $Y = F_Y^{-1}(U)$ ### Proof $F_Y(y) = P(Y \le y) = P(F_Y^{-1}(U) \le y) = P(U \le F_Y(y)) = F_Y(y)$ ### Other Methods 1. **Rejection Method** 2. **ziggurat Algorithm** ### Generating Gaussian Random Variables * **Box-Muller Method**: * Generate $U_1, U_2 \overset{iid}{\sim} Unif(0,1)$ * Let $$ \begin{aligned} &Z_1 = \sqrt{-2\ln(U_1)}\cos(2\pi U_2)\\ &Z_2 = \sqrt{-2\ln(U_1)}\sin(2\pi U_2) \end{aligned} $$ Then $Z_1, Z_2 \overset{iid}{\sim} N(0,1)$ ### Generating Poisson Random Variables * If $X \sim Poisson(\lambda)$, then $$ P(X=k) = e^{-\lambda}\frac{\lambda^k}{k!},\quad k=0,1,... $$ * **Fact**: If $X \sim Poisson(\lambda)$, then $X$ is equal to the number of arrivals in an interval of length 1, where the interarrival times are iid $\sim Exp(\lambda)$ ### Algorithm 1. $i \leftarrow 0$, $p \leftarrow e^{-\lambda}$, $F \leftarrow p$ 2. Generate $U \sim Unif(0,1)$ 3. While $U > F$ * $i \leftarrow i+1$ * $p \leftarrow p \cdot \frac{\lambda}{i}$ * $F \leftarrow F + p$ 4. Return $X = i$