Two Types of Random VariablesA random variable Show
x\text{x} , and its distribution, can be discrete or continuous. Learning Objectives Contrast discrete and continuous variables Key TakeawaysKey Points
Key Terms
Random Variables In probability and statistics, a randomvariable is a variable whose value is subject to variations due to chance (i.e. randomness, in a mathematical sense). As opposed to other mathematical variables, a random variable conceptually does not have a single, fixed value (even if unknown); rather, it can take on a
set of possible different values, each with an associated probability. Discrete Random VariablesDiscrete random variables can take on either a finite or at most a countably infinite set of discrete values (for example, the integers). Their probability distribution is given by a probability mass function which directly maps each value of the random variable to a probability. For example, the value of x1\text{x}_1 takes on the probability p1\text{p}_1 , the value of x2\text{x}_2 takes on the probability p2\text{p}_2 , and so on. The probabilities pi\text{p}_\text{i} must satisfy two requirements: every probability pi\text{p}_\text{i} is a number between 0 and 1, and the sum of all the probabilities is 1. ( p1+p2+⋯+pk=1\text{p}_1+\text{p}_2+\dots + \text{p}_\text{k} = 1 ) Discrete Probability Disrtibution: This shows the probability mass function of a discrete probability distribution. The probabilities of the singletons {1}, {3}, and {7} are respectively 0.2, 0.5, 0.3. A set not containing any of these points has probability zero. Examples of discrete random variables include the values obtained from rolling a die and the grades received on a test out of 100. Continuous Random VariablesContinuous random variables, on the other hand, take on values that vary continuously within one or more real intervals, and have a cumulative distribution function (CDF) that is absolutely continuous. 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.  Probability Density Function: The image shows the probability density function (pdf) of the normal distribution, also called Gaussian or "bell curve", the most important continuous random distribution. 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 VariablesProbability distributions for discrete random variables can be displayed as a formula, in a table, or in a graph. Learning Objectives Give examples of discrete random variables Key TakeawaysKey Points
Key Terms
A discrete random variable x\text{x} has a countable number of possible values. The probability distribution of a discrete random variable x\text{x} lists the values and their probabilities, where value x1\text{x}_1 has probability p1\text{p}_1 , value x2\text{x}_2 has probability x2\text{x}_2 , and so on. Every probability pi\text{p}_\text{i} is a number between 0 and 1, and the sum of all the probabilities is equal to 1.
A discrete probability distribution can be described by a table, by a formula, or by a graph. For example, suppose that x\text{x} is a random variable that represents the number of people waiting at the line at a fast-food restaurant and it happens to only take the values 2, 3, or 5 with probabilities 210\frac{2}{10} , 310\frac{3}{10} , and 510\frac{5}{10} respectively. This can be expressed through the function f(x)=x10\text{f}(\text{x})= \frac{\text{x}}{10} , x=2,3,5\text{x}=2, 3, 5 or through the table below. Of the conditional probabilities of the event B\text{B} given that A 1\text{A}_1 is the case or that A2\text{A}_2 is the case, respectively. 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. Expected Values of Discrete Random VariablesThe expected value of a random variable is the weighted average of all possible values that this random variable can take on. Learning Objectives Calculate the expected value of a discrete random variable Key TakeawaysKey Points
Key Terms
Discrete Random VariableA discrete random variable X\text{X} has a countable number of possible values. The probability distribution of a discrete random variable X\text{X} lists the values and their probabilities, such that xi\text{x}_\text{i} has a probability of pi\text{p}_\text{i} . The probabilities p i\text{p}_\text{i} must satisfy two requirements:
Expected Value Definition 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. The weights used in computing this average are probabilities in the case of a discrete random variable. How To Calculate Expected ValueSuppose random variable X\text{X} can take value x1\text{x}_1 with probability p1\text{p}_1 , value x2\text{x}_2 with probability p2\text{p}_2 , and so on, up to value xi\text{x}_i with probability pi\text{p}_i . Then the expectation value of a random variable X\text{X} is defined as: E[X]=x1p1+x2p2+⋯+xipi\text{E}[\text{X}] = \text{x}_1\text{p}_1 + \text{x}_2\text{p}_2 + \dots + \text{x}_\text{i}\text{p}_\text{i} , which can also be written as: E[X]=∑xipi\text{E}[\text{X}] = \sum \text{x}_\text{i}\text{p}_\text{i} . xi\text{x}_\text{i} are equally likely (that is, p1=p2=⋯=pi\text{p}_1 = \text{p}_2 = \dots = \text{p}_\text{i} ), then the weighted average turns into the simple average. This is intuitive: the expected value of a random variable is the average of all values it can take; thus the expected value is what one expects to happen on average. If the outcomes xi\text{x}_\text{i} are not equally probable, then the simple average must be replaced with the weighted average, which takes into account the fact that some outcomes are more likely than the others. The intuition, however, remains the same: the expected value of X\text{X} is what one expects to happen on average. X \text{X} represent the outcome of a roll of a six-sided die. The possible values for X\text{X} are 1, 2, 3, 4, 5, and 6, all equally likely (each having the probability of 16\frac{1}{6} ). The expectation of X\text{X} is: E[X]=1x16+2x26+3x36+4x4 6+5x56+6x66=3.5\text{E}[\text{X}] = \frac{1\text{x}_1}{6} + \frac{2\text{x}_2}{6} + \frac{3\text{x}_3}{6} + \frac{4\text{x}_4}{6} + \frac{5\text{x}_5}{6} + \frac{6\text{x}_6}{6} = 3.5 . In this case, since all outcomes are equally likely, we could have simply averaged the numbers together: 1+2+3+4+5+66=3.5\frac{1+2+3+4+5+6}{6} = 3.5 . Average Dice Value Against Number of Rolls: An illustration of the convergence of sequence averages of rolls of a die to the expected value of 3.5 as the number of rolls (trials) grows. Licenses and AttributionsCC licensed content, Shared previously
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What is a type of variable that can take infinite number on the value that can occur within a population Brainly?Continuous variables
A variable is said to be continuous if it can assume an infinite number of real values within a given interval.
What type of variable that assumes an infinite number of values in an interval between two specific values Brainly?A continuous random variable is one which takes an infinite number of possible values.
Can take the infinite number on the value that can occur within a population?Continuous random variables can represent any value within a specified range or interval and can take on an infinite number of possible values.
What variables that can take on a finite number of distinct values?A random variable is called discrete if it has either a finite or a countable number of possible values. A random variable is called continuous if its possible values contain a whole interval of numbers.
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What is type of variable that can take infinite number on the value that can occur within a population Brainly?
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