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Difference Between Discrete and Continuous Probability Distributions

Discrete vs Continuous Probability Distributions

Statistical experiments are random experiments that can be repeated indefinitely with a known set of outcomes. A variable is said to be a random variable if it is an outcome of a statistical experiment. For example, consider a random experiment of flipping a coin twice; the possible outcomes are HH, HT, TH, and TT. Let the variable X be the number of heads in the experiment. Then, X can take the values 0, 1 or 2, and it is a random variable. Observe that there is a definite probability for each of the outcomes X = 0, X = 1, and X = 2.

Thus, a function can be defined from the set of possible outcomes to the set of real numbers in such a way that ƒ(x) = P(X=x) (the probability of X being equal to x) for each possible outcome x. This particular function f is called the probability mass/density function of the random variable X. Now the probability mass function of X, in this particular example, can be written as ƒ(0) = 0.25, ƒ(1) = 0.5, ƒ(2) = 0.25.

Also, a function called cumulative distribution function (F) can be defined from the set of real numbers to the set of real numbers as F(x) = P(X ≤x) (the probability of X being less than or equal to x) for each possible outcome x. Now the cumulative distribution function of X, in this particular example, can be written as F(a) = 0, if a<0; F(a) = 0.25, if 0≤a<1; F(a) = 0.75, if 1≤a<2; F(a) = 1, if a≥2.

What is a discrete probability distribution?

If the random variable associated with the probability distribution is discrete, then such a probability distribution is called discrete. Such a distribution is specified by a probability mass function (ƒ). The example given above is an example of such a distribution since the random variable X can have only a finite number of values. Common examples of discrete probability distributions are binomial distribution, Poisson distribution, Hyper-geometric distribution and multinomial distribution. As seen from the example, cumulative distribution function (F) is a step function and ∑ ƒ(x) = 1.

What is a continuous probability distribution?

If the random variable associated with the probability distribution is continuous, then such a probability distribution is said to be continuous. Such a distribution is defined using a cumulative distribution function (F). Then it is observed that the probability density function ƒ(x) = dF(x)/dx and that ƒ(x) dx = 1. Normal distribution, student t distribution, chi squared distribution, and F distribution are common examples for continuous probability distributions.

What is the difference between a discrete probability distribution and a continuous probability distribution?

• In discrete probability distributions, the random variable associated with it is discrete, whereas in continuous probability distributions, the random variable is continuous.

• Continuous probability distributions are usually introduced using probability density functions, but discrete probability distributions are introduced using probability mass functions.

• The frequency plot of a discrete probability distribution is not continuous, but it is continuous when the distribution is continuous.

• The probability that a continuous random variable will assume a particular value is zero, but it is not the case in discrete random variables.


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Related posts:

  1. Difference Between Discrete and Continuous Variables
  2. Difference Between Discrete Function and Continuous Function
  3. Difference Between Discrete and Continuous Data
  4. Difference Between Poisson Distribution and Normal Distribution
  5. Difference Between Probability and Possibility

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