ch 5 continuous random variables
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CONTINUOUS RANDOM VARIABLES
CHAPTER 5
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RANDOM VARIABLES• Random Variable: A random variable describes the outcomes of a statistical experiment
• Random variables can be of two types
• Discrete random variables: can only take a finite or countable number of outcomes
• Example: number of patients with a particular disease
• Continuous random variables: take values within a specified interval or continuum
• Example: height or weight
• The values of a random variable can vary with each repetition of an experiment
• We will use capitol letters to denote random variables and lower case letters to denote the value of a random variable
• If X is a random variable, then x is written in words, and x is given as a number
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CONTINUOUS RANDOM VARIABLE EXAMPLE
• Let X = the number of minutes until the next earthquake in California
• Is X a continuous random variable?
• The sample space S = { ??? }
• X is in words and x is a number
• The x values are not countable (continuous or infinite) outcomes
• Because you cannot count the possible values that X can take on and the outcomes are random, X is a continuous random variable
• Other examples of continuous random variables?
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PROBABILITY DISTRIBUTION FUNCTION• Just like a discrete random
variable, every continuous random variable has an associated probability distribution function
• The graph of a continuous probability distribution is a curve
• Probability is represented as area under the curve
• Example: what is the probability that X is between 8 and 11?
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PROBABILITY DISTRIBUTION FUNCTION• In the discrete case, we could count all of the possible outcomes and assign each one a probability; in
the continuous case we CANNOT (why?)
• Instead of counting outcomes, we will be working now with area underneath the curves to solve probability problems
• The entire area under the curve is equal to one
• Probability will be found for intervals of X rather than for individual X values
• P(c < X < d) is the probability the random variable X is in the interval between values c and d
• For EVERY continuous random variable, the probability that P(X = c) is ALWAYS equal to 0, i.e. P(X = c) = 0 (why?)
• P(c < X < d) is the same as P(c ≤ X ≤ d) because probability is equal to area
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PROBABILITY DISTRIBUTION FUNCTION• We will find the area that represents the probability using
• Geometry
• Formulas
• Technology
• Probability Tables
• In general, calculus is needed to find the area under the curve for many probability distributions (in particular taking integrals)
• We will NOT be using calculus in this class!
• For continuous probability distributions functions,
PROBABILITY = AREA
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CONTINUOUS PROBABILITY FUNCTION
• Notation: We will use f(x) to denote the probability density function (pdf)
• You may think of f(x) as a function
• We define f(x) so that the area between it and the x-axis is equal to a probability
• Since the maximum probability is one, the maximum area is also one
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EXAMPLE 1
• Consider the pdf f(x) = for 0 ≤ x ≤ 20
• The graph of f(x) is a horizontal line
• What is the probability that 5 ≤ x ≤ 10?
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EXAMPLE 1 CONTINUED
• What is the probability that 5 ≤ x ≤ 10?
• P(5 ≤ x ≤ 10) = (10 – 5) = 0.25
• Recall:
Area of a rectangle = (base)(height)
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THE UNIFORM DISTRIBUTION
• The uniform distribution is a continuous probability distribution
• The uniform distribution describes data where the events are equally likely to occur
• Example: Age could be a continuous random variable that follows a uniform distribution
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THE UNIFORM DISTRIBUTION
• Let X be a uniform random variable
• X ~ U(a, b)
• The uniform probability density function is
• a = the lowest value of x
• b = the highest value of x
• The probability that c ≤ X ≤ d is
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EXAMPLE 2
• Let X ~ U(0, 23)
• Find the probability that 2 < X < 18
• P(2 < X < 18) =
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EXAMPLE 2 CONTINUED
• Let X ~ U(0, 23)
• Find the probability that X > 12 given that X > 8
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EXPECTED VALUE (MEAN) AND STANDARD DEVIATION
• Let X ~ U (a, b) then:• The expected value of X is
• The standard deviation of X is
• Example: Let X ~ U(0, 23), then
• And
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PERCENTILES FOR UNIFORM RANDOM VARIABLES
• Recall: The Pth percentile divides the data between the lower P% and the upper (100 – P)% of the data
• P% of data values are less than (or equal to) the Pth percentile
• How can we find the value k corresponding to the Pth percentile?
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PERCENTILES FOR RANDOM VARIABLES• How can we find the value k corresponding to the Pth percentile?
• We know that P(X ≤ k) = Pth percentile, so
Pth percentile
Rearranging terms we get
Pth percentile)
• Example: Let X ~ U(0, 23), what value corresponds to the 78th percentile?
So 17.94 is the 78th percentile
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OTHER CONTINUOUS PROBABILITY DISTRIBUTIONS• Some other well known continuous probability distributions
• The normal (Gaussian) distribution
• The exponential distribution
• The student-t distribution
• The χ2 distribution (Chi-squared – rhymes with Pie)
• The gamma distribution
• The beta distribution
• The wiebull distribution
• …any many more
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5.1 CONTINUOUS DISTRIBUTION
• Handout
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HOMEWORK
• Page 329: 72, 73, 74, 77, 80, 84
Continuous Random Variable Uniform Distribution Class Activity Homework