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Understanding Random Variables: Discrete vs. Continuous Types

Learn about discrete and continuous random variables, probability distributions, and graphing techniques through real-world examples. Explore common types like uniform and normal distributions, and master key concepts in probability theory.

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Understanding Random Variables: Discrete vs. Continuous Types

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  1. Section 6.1 Discrete and Continuous Random Variables

  2. Sample Spaces • In Chapter 6, we studied sample spaces. If we toss two coins, the sample space is {HH, HT, TH, or TT}. • We statisticians like numbers. So, let’s convert this sample space to numbers by counting the number of heads. Then the sample space is {0, 1, 2}.

  3. Random Variables • If we let X = the number of heads, then X is a random variable. • Each time we perform a trial, we’d get different values for X. • A random variable is a variable whose value is a numerical outcome of a random phenomenon.

  4. Discrete and Continuous Random Variables • There are two types of random variables: discrete and continuous. • Discrete random variables have a finite (countable) number of possible values. We use a histogram to graph a discrete random variable. • Continuous random variables take on all values in an interval of numbers. We use a density curve to graph continuous random variables.

  5. Discrete Random Variables • Since discrete random variables have a countable number of outcomes, we can list each outcome and its probability in a table. This is called a probability distribution. x represents the random variable, and p(x) is the probability a random variable occurs. Notice the probabilities are all between 0 and 1. The sum of the probabilities = 1. The rules of probability are still the same!

  6. Histogram for the coins

  7. Grades • The instructor of a large class gives 15% each of A’s and D’s, 30% each of B’s and C’s, and 10% F’s. Choose a student at random from the class. The student’s grade on a 4-point scale (A=4.0) is a random variable x from 0 - 4. • Write the probability distribution of x. • Find the probability that the student has an A or a B. • Find P(2 < x < 4). • Find P(2 ≤ x ≤ 4). • Graph the probability distribution and describe the graph.

  8. Toss 4 Coins • Let x = the number of Heads in 4 coin tosses. • List all the possible outcomes of four different tosses. • Write the probability distribution of x. • Graph the probability distribution. Describe the graph. • Find the probability of tossing at least two heads. • Find the probability of tossing at least one head.

  9. Continuous Random Variables • Continuous Random Variables are over an interval, in which we cannot count all of the possible outcomes. • To find probabilities of continuous random variables, we use the area under the density curve. • Because there are infinitely many possibilities, we cannot assign probabilities to each value of x. • Recall that the area under a density curve has a total area of 1. It is always positive. Sound familiar???

  10. Common Types of Continuous Random Variables • Uniform Distribution – Has the same height throughout • Choose a number at random between 0 and 1. • Find P(0.3 ≤ X ≤ 0.7) • Find P(X ≤ 0.5) • Find P(X > 0.8) • Find P(X ≤ 0.5 or X > 0.8) • Note: Be sure that the area under the curve is 1. This will help you find the height. • Note: < or ≤ are essentially the same with continuous random variables, as an individual number has no area.

  11. Common Types of Continuous Random Variables • Normal Distribution • Suppose X is distributed normally with a mean o f 32 and a standard deviation of 2. • Find P(X > 35). • Find P(31 ≤ X ≤ 35).

  12. More on the Normal Distribution • An opinion poll asks an SRS of 1500 American adults what they consider to be the most serious problem facing our schools. Suppose that if we could ask ALL adults this question, 30% would say “drugs.” This random variable is denoted: ~ N(0.3, 0.0118), where is the predicted probability. Find the probability that the poll results differ from the truth about the population by more than two percentage points?

  13. Key Point to Remember • All continuous probability distributions assign probability 0 to every individual outcome. • So, P(x = 0.8) in a uniform distribution is 0. • P(0.5 < x < 0.8) = P(0.5 ≤ X ≤ 0.8) • We can ignore the distinction between < and ≤ in continuous random variables. • THE SAME IS NOT TRUE FOR DISCRETE RANDOM VARIABLES!!!

  14. Homework Chapter 6 #2, 4, 6, 8, 21, 24

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