Probability Distribution – Uniform

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Probability Distribution – Uniform: Probability Distribution is defined as a mathematical function which provides probabilities of occurrence of different possible outcomes in any given experiment. Probability Distributions can be divided into two categories:

  1. Discrete Probability Distribution: If the probabilities are defined on a discrete random variable, one which can only take a discrete set of values, then the distribution is said to be a discrete probability distribution. For example, the event of rolling a die can be represented by a discrete random variable with the probability distribution being such that each event has a probability equal to 1/6
  2. Continuous Probability Distribution: If the probabilities are defined on a continuous random variable, one which can take any value between two numbers, then the distribution is said to be a continuous probability distribution. For example, the temperature throughout a given day can be represented by a continuous random variable and the corresponding probability distribution is said to be continuous.

There is one very import aspect regarding Continuous probability distribution which needs to be understood well. Continuous probability distributions can only be defined for a range; continuous probability is equal to 0 for a given point. For example, the probability of receiving two inches of rain in June is zero because two inches is a single point in an infinite range of possible values. On the other hand, the probability of the amount of rain being between 1.99999999 and 2.00000001 inches has some positive value. In the case of continuous distributions, P(x1 <= X <= x2) = P(x1 < X < x2) because p{x1) = p{x2) = 0.

 

Cumulative Distribution Function: A cumulative distribution function (cdf), or simply distribution function, defines the probability that a random variable, X, takes on a value equal to or less than a specific value, x. It represents the sum, or cumulative value, of the Probabilities for the outcomes up to and including a specified outcome. The cumulative distribution function for a random variable, X, may be expressed as F(x) = P(X <= x).

For Example, Consider the probability function X = {1, 2, 3, 4}, p(x) = x / 10. For this distribution, F(3) = 0.6 = 0.1 + 0.2 + 0.3, and F(4) = 1 = 0.1 + 0.2 + 0.3 + 0.4.

 

Discrete Probability Distribution: A discrete uniform random variable is one for which the probabilities for all possible outcomes for a discrete random variable are equal. For example, consider the discrete uniform probability distribution defined as X = {1, 2, 3, 4, 5}, p(x) = 0.2. Cumulative Uniform Distribution for nth outcome is simple n times the probability of an event i.e. F(X) = n P(X)

Where F(X) is Cumulative Distribution and P(X) is probability of an event. So for above given example, F(3) = 0.6 = 0.1 + 0.2 + 0.3, and F(4) = 1 = 0.1 + 0.2 + 0.3 + 0.4.

Continuous Probability Distribution: The continuous uniform distribution is defined over a range that spans between some lower limit, a, and some upper limit, b, which serve as the parameters of the distribution. Outcomes can only occur between a and b, and since we are dealing with a continuous distribution, even if x lies within range i.e. a < x < b even then probability for single point is equal to zero i.e. P(X = x) = 0.

Uniform Probability Distribution Function is defined by the following formula:

UNI_DIS_1

UNI_DIS_2

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