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A comprehensive overview of continuous probability distributions, focusing on the normal distribution and its key properties. It explains the concept of probability density functions, the relationship between standard deviation and the shape of the normal curve, and the importance of the standard normal distribution. The document also includes examples and exercises to illustrate the concepts and their applications.
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TOPICS: Continous Probability Distribution:
A continuous probability distribution describes the probabilities of the possible values of a continuous random variable. Unlike a discrete random variable, which can take on only specific values (like 1, 2, 3, etc.), a continuous random variable can take on any value within a range.
Discrete vs. Continuous Probability Distributions Discrete Probability Distribution Values are specific points (e.g., 0, 1, 2…). Probability function: Probability Mass Function (PMF) Conditions:
Application in Continous Probability Distribution: (^0 2 5 ) The equation y=f(x) is a way to describe a relationship between two variables: x (input) and y (output).
Continuous vs. Discrete Probability Functions
Probability Density Function (PDF): The PDF is the function f(x) that describes the distribution. It defines the "density" of probabilities at each point within the range of the random variable. By itself, the PDF doesn’t give a probability for a specific value but shows how probability is distributed across the values. To get an actual probability for a range, we integrate the PDF over that range.
The equation y=f(x) is a way to describe a relationship between two variables: x (input) and y (output). Density Variable
Estimating Time Spent on a Task Imagine that a company tracks the time employees spend completing a particular task. Analysis shows that the time it takes to complete the task is uniformly distributed between 0 and 12 minutes. Problem The company wants to know the probability that an employee will take between 2 and 8 minutes to complete the task. Since time is continuously distributed, we can model it with a uniform continuous probability distribution from 0 to 12 minutes, where every time interval is equally likely. Example: Uniform Distribution
Estimating Time Spent on a Task Imagine that a company tracks the time employees spend completing a particular task. Analysis shows that the time it takes to complete the task is uniformly distributed between 0 and 12 minutes. Problem The company wants to know the probability that an employee will take between 2 and 8 minutes to complete the task. Since time is continuously distributed, we can model it with a uniform continuous probability distribution from 0 to 12 minutes, where every time interval is equally likely.
Example: JollyMc Yogurt customers are charged for the amount of Fruit topings they take. Sampling suggests that the amount of topings taken is uniformly distributed between 5 ounces and 15 ounces per customer. f(x)=1/10 for 5 < x < 15 = 0 elsewhere where: x = Yogurt cup filling weight Expected Value of x E(x) = (a + b)/ = (5+15)/ 10 Variance of x Var(x) = (b-a)2/12 = (15-5)2/ = 8.
Example: JollyMc Yogurt customers are charged for the amount of Fruit topings they take. Sampling suggests that the amount of topings taken is uniformly distributed between 5 ounces and 15 ounces per customer. f(x)=1/10 for 5 < x < 15 = 0 elsewhere where: x = Yogurt cup filling weight To find the probability that the amount of topping 𝑥 is between 12 and 15 ounces, we calculate:
Example: JollyMc Yogurt customers are charged for the amount of Fruit topings they take. Sampling suggests that the amount of topings taken is uniformly distributed between 5 ounces and 15 ounces per customer. f(x)=1/10 for 5 < x < 15 = 0 elsewhere where: x = Yogurt cup filling weight To find the probability that the amount of topping 𝑥 is between 12 and 15 ounces, we calculate:
Example: JollyMc Yogurt customers are charged for the amount of Fruit topings they take. Sampling suggests that the amount of topings taken is uniformly distributed between 5 ounces and 15 ounces per customer. f(x)=1/10 for 5 < x < 15 = 0 elsewhere where: x = Yogurt cup filling weight To find the probability that the amount of topping 𝑥 is between 12 and 15 ounces, we calculate:
