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Continuous Random Variables
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Continuous Random Variables

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Continuous Random Variables

Continuous Random Variables

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  • 1. 1.7 Continuous Random Variables
  • 2. Continuous Random Variables
    Suppose we are interested in the probability that a given random variable will take on a value on the interval from a to b where a and b are constants with a  b. First, we divide the interval from a to b into n equal subintervals of width x containing respectively the points x1, x2, … , xn.
    Suppose that the probability that the random variable will take on a value in subinterval containing xi is given by f(xi)x. Then the probability that the random variable will take on a value in the interval from a to b is given by
  • 3. Continuous Random Variables(cont’d)
    If f is an integrable function defined for all values of the
    random variable, the probability that the value of the
    random variables falls between a and b is defined by
    letting x  0 as
    Note: The value of f(x) does not give the probability that the
    corresponding random variable takes on the values x; in the
    continuous case, probabilities are given by integrals not by the
    values f(x).
  • 4. Continuous Random Variables(cont’d)
    f(x)
    P(a  X  b)
    a
    b
    x
    Figure: Probability as area under f
  • 5. Continuous Random Variables(cont’d)
    The probability that a random variable takes on value x, i.e.
    Thus, in the continuous case probabilities associated with
    individual points are always zero. Consequently,
  • 6. Continuous Random Variables(cont’d)
    The function f is called probability density function or simply probability density.
    Characteristics of the probability density function f :
    1.
    for all x.
    2.
    F(x) represents the probability that a random variable with
    probability density f(x) takes on a value less than or equal to
    x and the corresponding function F is called the cumulative
    distribution function or simply distribution function of the
    random variable X.
  • 7. Continuous Random Variables(cont’d)
    Thus, for any value x,
    F (x) = P(X  x)
    is the area under the probability density function over the interval - to x. Mathematically,
    The probability that the random variable will take on a value
    on the interval from a to b is given by
    P(a  X  b) = F (b) - F (a)
  • 8. Continuous Random Variables(cont’d)
    According to the fundamental theorem of integral calculus it follows that
    wherever this derivative exists.
    F is non-decreasing function, F(-) = 0 and F() = 1.
    kth moment about the origin
  • 9. Continuous Random Variables(cont’d)
    Mean of a probability density:
    kth moment about the mean:
  • 10. Continuous Random Variables(cont’d)
    Variance of a probability density
     is referred to as the standard deviation