Lab 7

Author

YOUR NAME HERE

Remember, follow the instructions below and use R Markdown to create a pdf document with your code and answers to the following questions on Gradescope. You may find a template file by clicking “Code” in the top right corner of this page.

A. Random sampling in R

In your own words, explain the difference between dnorm(), pnorm(), qnorm(), and rnorm().
Suppose we simulate x <- runif(1). What is the distribution of qnorm(x)?
Suppose we simulate x <- rnorm(1). What is the distribution of pnorm(x)?

B. Gambler’s ruin

A and B are playing a coin flipping game. A starts with $n_a$ pennies and B starts with $n_b$ pennies. A coin is flipped repeatedly and if it comes up heads, B gives A a penny. If it comes up tails, A gives B a penny. The game ends when one player has no more pennies.

Write a function run_one_sim(seed, n_a, n_b) to simulate one game. Repeatedly use your code with different values of seed to estimate each player’s probability of winning when $n_a = n_b = 10$.
Use your function to estimate each player’s probability of winning when $n_a = 1,\ldots, 5$ and $n_b = 1,\ldots, 5$, testing every combination. Organize your results in a 5 by 5 matrix and print it out. What do you notice?

C. One-dimensional random walks

In this part, you will simulate a one-dimensional random walk. Suppose you are at the point $x$ at time $t$. At time $t+1$, the probability of moving forwards to $x+1$ is $p$ and the chance of moving backwards to $x-1$ is $1-p$. Assume that at time $t=1$, you are at $x_1=0$.

Write a function random_walk() that takes as input a numeric n_steps and a numeric $p$ and simulates n_steps steps of the one-dimensional random walk with forward probability $p$. You may have other input arguments if desired. The output should be a length vector of length n_steps starting with 0 where the $i$th entry represents the location of the random walker at time $t=i$. For example, random_walk(5, .5) may return the vector $(0, 1, 2, 1, 2)$.
Use your function to generate a random walk of 500 steps with probability $.55$ and generate a line graph with $t=1,\ldots, 500$ on the x-axis and $x_1,\ldots, x_{500}$ on the y-axis.
Use your function to generate two more random walks of 500 steps with probability $p$, where $p\sim \mathrm{Unif}(0, 1)$ and create a line graph with all three of your random walks, using different colors for each walk.

--- title: "Lab 7" author: "YOUR NAME HERE" format: html: embed-resources: true code-tools: true code-summary: "Code" --- Remember, **follow the instructions below and use R Markdown to create a pdf document with your code and answers to the following questions on Gradescope.** You may find a template file by clicking "Code" in the top right corner of this page. ## A. Random sampling in R 1. In your own words, explain the difference between `dnorm()`, `pnorm()`, `qnorm()`, and `rnorm()`. 2. Suppose we simulate `x <- runif(1)`. What is the distribution of `qnorm(x)`? 3. Suppose we simulate `x <- rnorm(1)`. What is the distribution of `pnorm(x)`? ## B. Gambler's ruin A and B are playing a coin flipping game. A starts with $n_a$ pennies and B starts with $n_b$ pennies. A coin is flipped repeatedly and if it comes up heads, B gives A a penny. If it comes up tails, A gives B a penny. The game ends when one player has no more pennies. 4. Write a function `run_one_sim(seed, n_a, n_b)` to simulate one game. Repeatedly use your code with different values of `seed` to estimate each player's probability of winning when $n_a = n_b = 10$. 5. Use your function to estimate each player's probability of winning when $n_a = 1,\ldots, 5$ and $n_b = 1,\ldots, 5$, testing every combination. Organize your results in a 5 by 5 matrix and print it out. What do you notice? ## C. One-dimensional random walks In this part, you will simulate a one-dimensional random walk. Suppose you are at the point $x$ at time $t$. At time $t+1$, the probability of moving forwards to $x+1$ is $p$ and the chance of moving backwards to $x-1$ is $1-p$. Assume that at time $t=1$, you are at $x_1=0$. 6. Write a function `random_walk()` that takes as input a numeric `n_steps` and a numeric $p$ and simulates `n_steps` steps of the one-dimensional random walk with forward probability $p$. You may have other input arguments if desired. The output should be a length vector of length `n_steps` starting with 0 where the $i$th entry represents the location of the random walker at time $t=i$. For example, `random_walk(5, .5)` may return the vector $(0, 1, 2, 1, 2)$. 7. Use your function to generate a random walk of 500 steps with probability $.55$ and generate a line graph with $t=1,\ldots, 500$ on the x-axis and $x_1,\ldots, x_{500}$ on the y-axis. 8. Use your function to generate two more random walks of 500 steps with probability $p$, where $p\sim \mathrm{Unif}(0, 1)$ and create a line graph with all three of your random walks, using different colors for each walk.