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Understanding Density Curves and Random Variables in Probability

This chapter focuses on density curves and random variables in probability, specifically exploring a piecewise linear density curve consisting of two segments. It begins by validating the density curve and calculating the proportions of observations within specified intervals. The chapter also distinguishes between continuous and discrete random variables, emphasizing how each is defined and measured. Practical examples illustrate the concept of discrete probability distributions, providing insight into finding probabilities for specific events and visualizing distributions through tables and histograms.

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Understanding Density Curves and Random Variables in Probability

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  1. Chapter 7 Day 1

  2. Warm – Up (Density Curves Review) • Consider the density curve that consists of two line segments. The first segment starts at the point (0,2) and ends at the point (0.4,1). The second segments begin at (0.4,1) and ends at (0.8,1). • 1. Verify that this is a valid density curve. • 2. Use the area under the density curve to find the proportions of observations within the given intervals. • A. P(0.6 < X < 0.8) • B. P(0 < X < 0.4) • C. P(0 ≤ X ≤ 0.2) • D. P(X = 0.5)

  3. Continuous vs. Discrete • Continuous is measured, infinite • Discrete is counted, finite

  4. Continuous vs. Discrete • Gliding down a slide • Pouring water • Length of a rope • Age of students • Seasonal rainfall • Distance of a race • Time playing a CD • Segment on a graph • Climbing up stairs • Stacking ice cubes • Number of knots • Number of your birthday • Rainy days • Number of participants • Points on a graph

  5. Random Variable • A random variable is a variable whose value is a numerical outcome of a random phenomenon.

  6. Discrete Random Variable • A discrete random variableX has a countable number of possible values. The probability distribution of a discrete random variable X lists the values and their probabilities.

  7. Discrete Random Variable Cont… • The probability p must satisfy two requirements: • 1. Every probability pi is a number between 0 and 1. • 2. The sum of the probabilities is 1. • Find the probability of an event by adding the probabilities pi of the particular values xi that make up that event.

  8. Example #2 from Homework • A couple plans to have three children. There are 8 possible arrangements of girls or boys. For example GGB means the first two children are girls and the third is a boy. All 8 arrangements are equally likely. • A. Write down all 8 arrangements. What is the probability of any one of these arrangements? • Let X be the number of girls the couple has. What is the probability that X = 2? • Starting from your work in A, find the distribution of X. That is, what values can X take, and what are the probabilities for each value. (make a table and histogram)

  9. Example #6 from Homework • Choose an American household at random and let the random variable X be the number of persons living in the household. If we ignore the few houses with more than 7 inhabitants, the probability distribution of X is as follows.

  10. Verify that this is a legitimate discrete probability distribution and draw a probability histogram to display it. • What is P(X ≥ 5)? • What is P(X > 5)? • What is P(2 < X < 4)? • What is P(X ≠ 1)? • Write the event that a randomly chosen household contains more than two persons in terms of the random variable X. What is the probability of this event?

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