Create FREE 'HowTo' Videos with MyGuide

Continuous Probability



Pass Quiz and Get a Badge of Learning



Content "filtered", Please subscribe for FULL access.


Chapter 8 : Continuous Probability



Continuous Probability arrow_upward


  • A random variable is called continuous if it can assume all possible values in the possible range of the random variable.
  • Probability Distribution of random variable is known as continuous probability distribution.
  • Examples:
    • Thickness of an item.
    • Time required to complete a task.
    • Temperature of a solution.
    • Height, in inches.
  • Continuous Probability Distribution includes the following:

  • Probability of the Random Variable arrow_upward


  • The Probability of the Random Variable assuming a value within some given interval from x1 to x2 is defined to be the area under the graph of the probability density function between x1 and x2 .
  • Probability is the area under the curve.

  • Uniform Distribution arrow_upward


  • A random variable is uniformly distributed whenever the probability is proportional to the interval’s length.
  • The uniform probability density function is given below:
  •    

  • Where,
    • f(x) = Value of the density function at any x value.
    • a = Minimum value of x.
    • b = Maximum value of x.
  • The uniform distribution is a probability distribution that has equal probabilities for all possible outcomes of the random variable.
  • Expected Value (mean) of x.
  •    

  • Variance of x.
  •    


    Normal Distribution arrow_upward


  • The Normal Probability Distribution is the most important distribution for describing a continuous random variable.
  • It is widely used in statistical inference:
    • Height of the people.
    • Amount of rainfall.

  • Properties of Normal Distribution:

  • Bell Shaped.
  • Symmetrical.
  • Mean = Median = Mode.
  • Location is determined by the mean, .
  • Spread is determined by the standard deviation, .
  • The random variable has an infinite theoretical range: .
    • Changing  shifts the distribution left or right.
    • Changing  increases or decreases the spread.


    Characteristics of the Normal Distribution:

  • The entire family of normal probability distributions is defined by its mean m and its standard deviations.
  • The distribution is symmetric if its Skewness measure is zero.
  • The highest point on the normal curve is at the mean, which is also the median and mode.
  • The mean can be any numerical value:  negative, zero, or positive.
  • The standard deviation determines the width of the curve: larger values result in wider, flatter curves.
  • Probabilities for the normal random variable are given by areas under the curve.
    • The total area under the curve is 1 (.5 to the left of the mean and .5 to the right)

  • For most data sets:
    • 68% of the values of a normal random variable are within 1 Standard Deviation (SD) of its mean.

    • 95% of the values of a normal random variable are within 2 SD of its mean.
    • 99.7% of the values of a normal random variable are within (±) 3 SD of its mean.


    Normal Distribution: Formula arrow_upward


  • Where, 
    •  Value of the density function at any  value.
    •  The mathematical constant approximated by 2.71828.
    • The mathematical constant approximated by 3.14159.
    •  The population mean.
    •  The population standard deviation.
    •  Any value of the continuous variable.

    Standardized Normal Distribution arrow_upward


  • A random variable having a normal distribution with a mean of 0 and a standard deviation of 1 is said to have a Standard Normal Probability Distribution.
  • Any normal distribution (with any mean and standard deviation combination) can be transformed into the standardized normal distribution.
    • Also known as the  distribution.
    • Mean is 0.
    • Standard Deviation is 1.
    • Values above the mean have positive Z-values.
    • Values below the mean have negative Z-values.

        

  • We can think ofz as a measure of the number of standard deviations x is from .
  • We use the above equation to convert normal distribution into standard normal distribution.
  •  


    Standard Normal Distribution: Formula arrow_upward


  • Where,
    •  Any value of the continuous variable.
    •  Value of the density function at any  value.
    •  The mathematical constant approximated by 2.71828.
    • The mathematical constant approximated by 3.14159.

    Exponential Distribution arrow_upward


  • It is used to model the length of time between two occurrences of an event (the time between arrivals).
  • Example:
  • Time between transactions at an ATM Machine.


  • Thank You from Kimavi arrow_upward


  • Please email us at Admin@Kimavi.com and help us improve this tutorial.


  • Mark as Complete => Receive a Certificate in Statistics


    Kimavi Logo

    Terms and conditions, privacy and cookie policy | Facebook | YouTube | TheCodex.Me | Email Kimavi


    Kimavi - A Video Learning Library { Learning is Earning }

    Get Ad Free Learning with Progress Report, Tutor Help, and Certificate of Learning for only $10 a month



    All videos on this site created using MyGuide.

    Create FREE HowTo videos with MyGuide.