Continuous Probability Distribution

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In the beginning of the course we looked at the difference between discrete and continuous data. The last section explored working with discrete data, specifically, the distributions of discrete data. In this lesson we're again looking at the distributions but now in terms of continuous data.

A continuous probability distribution is made up of continuous variables.


                                                       Continuous probability distribution of men’s heights.

A probability distribution in which the random variable X can take on any value (is continuous). Because there are infinite values that X could assume, the probability of X taking on any one specific value is zero. Therefore we often speak in ranges of values (p(X>0) = .50). The normal distribution is one example of a continuous distribution.

Examples of continuous data include...

  • the amount of rainfall in inches in a year for a city.
  • the weight of a newborn baby.
  • the height of a randomly selected student.

Expected value and Variance of a Continuous Random Variable

The expected value and the variance have the same meaning (but different equations) as they did for the discrete random variables.

Expected Value (or mean) of a Continuous Random Variable :The expected value (or mean) of a continuous random variable is denoted by μ=E(Y).

Variance of a Continuous Random Variable:The variance of a continuous random variable is denoted by σ2=Var(Y).

Standard Deviation of a Continuous Random Variable:The standard deviation of a continuous random variable is denoted by σ=Var(Y)

Notice the equations are not provided for the three parameters above. Therefore, for the continuous case, you will not be asked to find these values by hand.

Types of Continuous Probability Distribution

The normal distribution is the “go to” distribution for many reasons, including that it can be used the approximate the binomial distribution, as well as the hypergeometric distribution and Poisson distribution.
Other continuous distributions that are common in statistics include

  • Beta distribution
  • Cauchy distribution
  • Exponential distribution
  • Gamma distribution
  • Logistic distribution
  • Weibull distribution