Understanding Continuous Random Variables in Probability Lab

STAT 3090 S ECTION 6 L AB S PRING 2024 C ONTINUOUS R ANDOM V ARIABLES 1 N AME : Matt Schmidt 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. 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 (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) = ∫ 𝑓(𝑥)𝑑𝑥 = 0.6321 20 𝑛 (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(15 ≤ 𝑥 ≤ 30) = ∫ 𝑓(𝑥)𝑑𝑥 = 0.2492 30 15

STAT 3090 S ECTION 6 L AB S PRING 2024 C ONTINUOUS R ANDOM V ARIABLES 2 2. 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) F(x) is greater than 0, and the sum of all the probabilities is equal to 1 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) = ∫ 𝑓(𝑥)𝑑𝑥 = 0.42 9 3 3. Calculate the expected loss due to a fire. Include the correct symbol in your answer.(5 pts) ?[𝑥] = ∫ 𝑥 ∗ ?(𝑥)𝑑𝑥 = ∫ 𝑥 ∗ 0.005(20 − 𝑥)𝑑𝑥 = 6.667 ?𝑖??𝑖?? 20 0 20 0 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) 𝑉??(𝑥) = ∫ (? − 20 3 ) 20 0 2 * f(x)dx = 200/9 = 22.222 million dollars

STAT 3090 S ECTION 6 L AB S PRING 2024 C ONTINUOUS R ANDOM V ARIABLES 3 5. What is the standard deviation for the loss due to a fire? Include the correct symbol with your answer. (5 pts) √ 200 9 = 4.714 ?𝑖??𝑖?? ****************************************************************************** Part 2 4. A Relative of the Normal 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 ( 𝜒 2 ) 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 (−∞ < 𝑋 < ∞) . (1 pt) c) The Standard Normal distribution has the parameters 𝜇 = 0 and 𝜎 = 1 (2 pts) d) If Z is a random variable that follows a Standard Normal distribution, then (−∞ < ? < ∞) . (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 Normal Data . Highlight the Normal 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)

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STAT 3090 S ECTION 6 L AB S PRING 2024 C ONTINUOUS R ANDOM V ARIABLES 4 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 Normal data in our JMP file. Right click the Column header and select Formula . Click the 𝑥 𝑦 button. Then, click Normal 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)

STAT 3090 S ECTION 6 L AB S PRING 2024 C ONTINUOUS R ANDOM V ARIABLES 5 iii. Your new data distribution is known as the Chi-Squared ( 𝜒 2 ) 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 distribution of the standard normal variable is symmetric. The chi-squared graph is very skewed to the left. The mean of the normal is much higher than the mean of the squared graph. 5. 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.

STAT 3090 S ECTION 6 L AB S PRING 2024 C ONTINUOUS R ANDOM V ARIABLES 6 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 𝝌 ? 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) This graph follows the squared graph much more as the distribution os skewed to the left. 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 𝜒 2 cdf. Enter the lower bound, upper bound and degrees of freedom (which w ill 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. The lower bound is 0 and the upper bound is 1

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STAT 3090 S ECTION 6 L AB S PRING 2024 C ONTINUOUS R ANDOM V ARIABLES 7 𝑷(𝒙 ? ≤ ?. ?) = ?. ?𝟒𝟖 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) The lower bound is 0 and the upper bound is 1 𝑷(𝒙 ? ≤ ?. ?) = ?. ??? 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) 𝑃(0.1 ≤ 𝑥 2 ≤ 1.5) = 𝑃(𝑥 2 ≤ 1.5) − 𝑃(𝑥 2 ≤ 0.1 ) = 0.531

Sec 6 Lab

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