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Ch 5.1 Continuous Random Variable and Density Curve

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  • Ch 5.1 Continuous random variable

    A) Density Curve

    Probability of a Continuous Random Variable X is  defined by its Probability Density Function(pdf) or density curve: \( f(x) \) so that  

    - Area under the density curve corresponds to probability or relative frequency (percent).

    Density curve

    - Total area under the density curve is equal to 1

    Density curve

    - the graph is always above x-axis.

       Probability = Area = Percent

    Two important continuous Probability Distributions

    1) Uniform Distribution – The probability of X is equally likely to occur. Histogram of sample data usually bars of similar heights.  There is lowest and highest value of X.

    Ex1.  X is modeled by Uniform Distribution for

      lowest 2 and highest 8.8.

    Probability that X is between 3 and 6 is the shaded area under the density curve.

    Density Curve

    b) Normal Distribution

    Density curve bell curve

    Probability that X is between a and b =

    the area under the bell curve for x = a and x = b.



    Left area bell curve 

    Shaded left area =

    probability that x is less than a.


    Right area bell curve

    Shaded right area  =

    probability that x is greater than a.



    Notation and property of probability of Continuous random variable X.

    Probability that X = a:  P(X = a) = 0

    Probability that X is between a and b:

    P( a < X < b) or  P(a ≤ X ≤ b)

    Probability that X is less than a: P( X < a) = P( X ≤ a)

    Probability that X is greater than a: P(X > a) = P(x ≥ a)


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