# 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%
Example
• 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) 0 0.25 0.25 1 0.50 0.75 2 0.25 1.00