Discrete and Continuous Random Variables:. A variable is a quantity whose value changes. A discrete variable is a variable whose value is obtained by counting. Examples : number of students present. A continuous variable is a variable whose value is obtained by measuring. Examples : height of students in class. A random variable is a variable whose value is a numerical outcome of a random phenomenon. A discrete random variable X has a countable number of possible values.
Example : Let X represent the sum of two dice. Then the probability distribution of X is as follows:. To graph the probability distribution of a discrete random variable, construct a probability histogram. As a result, the random variable has an uncountable infinite number of possible values, all of which have probability 0, though ranges of such values can have nonzero probability. The resulting probability distribution of the random variable can be described by a probability density, where the probability is found by taking the area under the curve.
As notated on the figure, the probabilities of intervals of values corresponds to the area under the curve. Selecting random numbers between 0 and 1 are examples of continuous random variables because there are an infinite number of possibilities. Probability distributions for discrete random variables can be displayed as a formula, in a table, or in a graph.
A discrete probability distribution can be described by a table, by a formula, or by a graph. Notice that these two representations are equivalent, and that this can be represented graphically as in the probability histogram below. Probability Histogram : This histogram displays the probabilities of each of the three discrete random variables. The formula, table, and probability histogram satisfy the following necessary conditions of discrete probability distributions:.
Sometimes, the discrete probability distribution is referred to as the probability mass function pmf. The probability mass function has the same purpose as the probability histogram, and displays specific probabilities for each discrete random variable. The only difference is how it looks graphically. Probability Mass Function : This shows the graph of a probability mass function. All the values of this function must be non-negative and sum up to 1.
Discrete Probability Distribution : This table shows the values of the discrete random variable can take on and their corresponding probabilities. The expected value of a random variable is the weighted average of all possible values that this random variable can take on. In probability theory, the expected value or expectation, mathematical expectation, EV, mean, or first moment of a random variable is the weighted average of all possible values that this random variable can take on.
For example, categorical predictors include gender, material type, and payment method. Discrete variable Discrete variables are numeric variables that have a countable number of values between any two values.
A discrete variable is always numeric. For example, the number of customer complaints or the number of flaws or defects.
Continuous variable Continuous variables are numeric variables that have an infinite number of values between any two values. For example, the length of a part or the date and time a payment is received.
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