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Probability Distribution

  • A Random Variable is a function, which assigns unique numerical values to all possible outcomes of a random experiment under fixed conditions. A random variable is not a variable but rather a function that maps events to numbers
  • Probability distribution describes the values and probabilities that a random event can take place. The values must cover all of the possible outcomes of the event, while the total probabilities must sum to exactly 1, or 100%
  • Suppose you flip a coin twice.
  • There are four possible outcomes: HH, HT, TH, and TT.
  • Let the variable X represent the number of Heads that result from this experiment
  • It can take on the values 0, 1, or 2
  • X is a random variable (its value is determined by the outcome of a statistical experiment)
  • A probability distribution is a table or an relation that links each outcome of a statistical experiment with its probability of occurrence
Number of heads (X) Probability P(X=x)
0 0.25
1 0.50
2 0.25


Continuous & Discrete Probability Distributions


  • If a variable can take on any value between two specified values, it is called a continuous variable
    • Otherwise, it is called a discrete variable
  • If a random variable is a discrete variable, its probability distribution is called a discrete probability
    • For example, tossing of a coin & noting the number of heads (random variable) can take a discrete value
    • Binomial probability distribution, Poisson probability distribution
  • If a random variable is a continuous variable, its probability distribution is called a continuous probability distribution
    • The probability that a continuous random variable will assume a particular value is zero
    • A continuous probability distribution cannot be expressed in tabular form
    • An equation or formula is used to describe a continuous probability distribution (called a probability density function or density function or PDF)
    • The area bounded by the curve of the density function and the x-axis is equal to 1, when computed over the domain of the variable
    • Normal probability distribution, Student's t distribution are examples of continuous probability distributions

Probability Distribution


  • Cumulative Probability is a rule or equation which describes the sum of all the probabilities till that observation
  • Take the previous example of flipping of coin twice. The following table gives the probability of occurrence of heads and the cumulative probability as well
  • The point to note here is that the cumulative probability of the first event is equal to the probability of that event
  • The cumulative probability of the last event is always 1

Number of heads (X)

Probability P(X=x)

Cumulative Probability: P(X < x)











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