Continuous Random Variables: Applying Calculus in Probability Lab

STAT 3090 S ECTIO N 6 L AB S PRI N G 2024 C O N TI N UOUS R A N DOM V ARIABLES N AME : P URPOSE : In this lab you will apply your understanding of calculus to solve various types of probability problems dealing with continuous random variables. O BJECTIVES : Upon successful completion of this assignment, you will be able to… • Calculate probabilities for a continuous random variable • Find the mean and variance/standard deviation for a continuous random variable. • Calculate probabilities for a normally distributed random variable using a table or calculator • Find the value of a normally distributed random variable given a probability • Simulate the Chi-Squared distribution and use it to solve problems. Part One 1. Webassign Section 6: #1 - #2 (9 pts) 2. The Red Iguana (8 pts) Dinner customers at the Red Iguana Restaurant often experience a long wait for a table. For a randomly selected customer who arrives at the restaurant between 6:00 PM and 7:00 PM, the waiting time (in minutes) is a continuous random variable such that 1

STAT 3090 S ECTIO N 6 L AB S PRI N G 2024 C O N TI N UOUS R A N DOM V ARIABLES (a) Suppose a dinner customer is randomly selected. What is the probability that the person must wait for a table at most 20 minutes? Show correct probability notation. (4 pts) P(X<20)= integral from 0 to 20 of (.05 e^-.05x)dx= .6321 (b) Suppose a dinner customer is randomly selected. What is the probability that the person must wait for a table between 15 and 30 minutes? Show correct probability notation. (4 pts) P(X<x<30)= intergral of 15 to 30 of (.05 e^-.05x)dx= .2492 3. Property Loss (25 pts) The loss (in million dollars) due to a fire in a commercial building is modeled by a random variable X with a probability density function of Use this information to answer the following questions. 1. Verify that we have a legitimate probability distribution. There are two conditions that need to be satisfied. You may sketch the graph to help you answer this question. (5 pts) All positive & all equal to 1 P(x<x<20)= integral of 0 to 20 of (.005 (20-x))dx =1 2

STAT 3090 S ECTIO N 6 L AB S PRI N G 2024 C O N TI N UOUS R A N DOM V ARIABLES 2. What is the probability that the loss due to a fire is between $3 million and $9 million dollars? Show correct probability notation. (5 pts) P(3<x<9)= intergral from 3 to 9 of (.005(20-x))dx= .42 3. Calculate the expected loss due to a fire. Include the correct symbol in your answer.(5 pts) Mean= integral from 0 to 20 of (x(.005(20-x)))dx= 20/30= 6.67 million dollars 4. What is the variance of the loss due to a fire in a commercial building? Include the correct symbol in your answer. (5 pts) Variance= intergral from 0 to 20 of (x-20/3)^2 * f(x)dx =200/9= 22.22 million dollars 5. What is the standard deviation for the loss due to a fire? Include the correct symbol with your answer. (5 pts) SD= square root of 22.22= 4.714 3

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STAT 3090 S ECTIO N 6 L AB S PRI N G 2024 C O N TI N UOUS R A N DOM V ARIABLES ****************************************************************************** Part 2 4. WebAssign: #3 – 15 (28 points) 5. A Relative of the N ormal Distribution (20 pts) Data File: JMP file S&P500_5_yr_alpha Preliminaries for problem 3: The Normal distribution is very important to statistics. There are many distributions that can be related to the Normal distribution. A relative of the Normal distribution that we will need later in the course is called the Chi Squared ( ) distribution. Let’s see how this distribution comes about and see if we can apply our knowledge of probability distributions in the problem 3. a) As you have learned, the Normal distribution has two parameters, the ___mean________ and the ____standard deviation___________________ (2pts) b) If X is a random variable that follows a Normal distribution, then negative infinity to infinity . (1 pt) c) The Standard Normal distribution has the parameters __0____ and ___1_____ (2 pts) d) If Z is a random variable that follows a Standard Normal distribution, then negative infinity to infinity . (1 pt) e) Suppose you take the random variable z from the Standard Normal distribution and square its value. What do you think will happen? Let’s try it. i. First create a histogram and summary statistics for a random sample of values from the Standard Normal distribution. You will find the sample in the data file JMP file S&P500_5_yr_alpha , in the column labeled N ormal Data . Highlight the N ormal Data column. Then, go to the Analyze tab in JMP, and click Distribution. After doing this, make sure to select the Stack choice for the histogram. Paste the histogram and tables here. (6 pts) χ 2 (_______ < X < _______) μ = σ = (______ < Z < ________) 4

STAT 3090 S ECTIO N 6 L AB S PRI N G 2024 C O N TI N UOUS R A N DOM V ARIABLES ii. Create a new column in JMP with the squared values of the sample from the Standard Normal distribution. To do this, double click the column to the right of the N ormal data in our JMP file. Right click the Column header and select Formula . Click the button. Then, click N ormal Data button where the column titles are listed. The default value for y is 2 so you do not need to make any other changes. Select OK . Create a histogram and summary statistics for this new data. Go to the Analyze tab in JMP, and click Distribution and use your new column . After doing this, make sure to select the Stack choice for the histogram. Paste the histogram and tables here. (6 pts) x y 5

STAT 3090 S ECTIO N 6 L AB S PRI N G 2024 C O N TI N UOUS R A N DOM V ARIABLES iii. Your new data distribution is known as the Chi-Squared ( ) distribution with degrees of freedom equal to 1 (we will learn more about degrees of freedom later). Write a sentence or two to compare the shape and means of these two distributions. In this problem it will be helpful for you to round the means to the nearest integer. You do not need to write a formal comparison of two distributions, simply compare the shapes and means. (2 pts) The first graph has a mean of 0 & the second has a mean of 1. The first graph is also more skewed left while the second is skewed right but they are both unimodal. 6. Mr. Morgan Plays the Market. (10 pts) a) You, a portfolio analyst at the mid-sized investment bank Morgan & Sons, have been tasked with analyzing one of the firm’s portfolios of US stocks. In particular, you would like to analyze just how well this portfolio outperformed the general market. In finance parlance, this is known as computing the alpha return. Due to various macroeconomic events, you will be analyzing the adjusted alpha-return. You will find the data for this analysis in the JMP file S&P500_5_yr_alpha. Your colleague Milton, a PhD economist, believes that the adjusted alpha-returns should have an approximate normal distribution. Create a histogram and summary tables for the adjusted alpha-return data. Paste the output below and comment on whether you believe that the data follows a normal distribution, or if it follows the distribution with degrees of freedom equal to 1 . You do not need to make a formal comparison of the distributions, simply write your comments. (4 pts) It has a mean of 1 so it follows x^2 χ 2 χ 2 6

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STAT 3090 S ECTIO N 6 L AB S PRI N G 2024 C O N TI N UOUS R A N DOM V ARIABLES b) Mr. Morgan, the Managing Partner of Morgan & Sons, is thinking about changing what stocks are in this portfolio. Before making any position decisions, he would like you to compute certain return probabilities dealing with the portfolio. For this problem, assume that adjusted alpha-returns have a Chi-Squared Distribution with degrees of freedom 1. i. The Board of Directors will fire Mr. Morgan (and give him an exceedingly generous golden parachute) if adjusted alpha-returns do not exceed 0.1. Find the probability that Mr. Morgan is fired. Round your answer to three decimals. (2 pts) Calculator Instructions: The Chi-Squared Distribution functions in your calculator work very similar to the Normal Distribution functions. Rather than selecting normalcdf, you will select cdf. Enter the lower bound, upper bound and degrees of freedom (which will be 1 in today’s problem). Note, the smallest value for a Chi-Squared distribution is 0 and the largest value is . To show your work write down the lower bound and upper bound you are using in the calculator. LB= 0 UB=1 .248 probability Mr. Morgan will be fired ii. Mr. Morgan will receive an exceedingly large bonus if adjusted alpha-returns exceed 1.5. What is the probability Mr. Morgan gets his bonus? Round your answer to three decimals. To show your work write down the lower bound and upper bound you are using. (2 pts) LB=1.5 UB=+infinity .221 probability he will get a raise iii. If Adjusted alpha-returns are between 0.1 and 1.5, no one on your team will receive their bonuses. What is the probability no one receives their bonus? Round your answer to three decimals. Show your work by either using the answers from above or write down the lower and upper bounds. (2 pts) LB=.1 UB=1.5 1-.221=.779 P(x<1.5) - P(x<.1)=.531 χ 2 + ∞ 7

STAT 3090 S ECTIO N 6 L AB S PRI N G 2024 C O N TI N UOUS R A N DOM V ARIABLES 8

Sec 6 Lab (1)

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