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LECTURE UNIT 4.3

LECTURE UNIT 4.3. Normal Random Variables and Normal Probability Distributions. Understanding Normal Distributions is Essential for the Successful Completion of this Course. Recall: Probability Distributions p(x) for a Discrete Random Variable. p(x) = Pr(X=x) Two properties

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LECTURE UNIT 4.3

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  1. LECTURE UNIT 4.3 Normal Random Variables and Normal Probability Distributions

  2. Understanding Normal Distributions is Essential for the Successful Completion of this Course

  3. Recall: Probability Distributions p(x) for a Discrete Random Variable • p(x) = Pr(X=x) • Two properties 1. 0  p(x)  1 for all values of x 2.  all x p(x) = 1

  4. The sum of all the areas is 1 p(5)=.246 is the area of the rectangle above 5 Graph of p(x); x binomial n=10 p=.5; p(0)+p(1)+ … +p(10)=1 Think of p(x) as the area of rectangle above x

  5. Recall: Continuous r. v. x • A continuous random variable can assume any value in an interval of the real line (test: no nearest neighbor to a particular value)

  6. Discrete random variable p(x): probability distribution function for a discrete random variable x Continuous random variable f(x): probability density function of a continuous random variable x Discrete rv: prob dist functionCont. rv: density function

  7. Binomial rv n=100 p=.5

  8. The graph of f(x) is a smooth curve f(x)

  9. Graphs of probability density functions f(x) • Probability density functions come in many shapes • The shape depends on the probability distribution of the continuous random variable that the density function represents

  10. Graphs of probability density functions f(x) f(x) f(x) f(x)

  11. P(a < X < b) Probabilities: area under graph of f(x) f(x) X b a P(a < X < b) = area under the density curve between a and b. P(X=a) = 0 P(a < x < b) = P(a < x < b)

  12. f(x)0 for all x the total area under the graph of f(x) = 1 0  p(x)  1  p(x)=1 The sum of all the areas is 1 Total area under curve =1 Properties of a probability density function f(x) Think of p(x) as the area of rectangle above x f(x) x

  13. 1. 0  p(x)  1 for all values of x 2. all x p(x) = 1 values of p(x) for a discrete rv X are probabilities: p(x) = Pr(X=x); 1. f(x)0 for all x 2. the total area under the graph of f(x) = 1 values of f(x) are not probabilities - it is areas under the graph of f(x) that are probabilities Important difference

  14. Next: normal random variables

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