Sec 6 Lab (1)

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Clemson University *

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3090

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Statistics

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Apr 3, 2024

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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 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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