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 |