Continuous Random Variable Probability

Probability Density Function. Calculating probabilities for continuous random variables requires a different approach from the methods used with discrete variables. If all the outcomes of a continuous random variable are displayed in a histogram, the interval widths become infinitely small.

In statistics and probability theory, a continuous random variable is a type of variable that can take any value within a given range. Unlike discrete random variables, which can only assume specific, separate values like the number of students in a class, continuous random variables can assume any value within an interval, making them ideal for modelling quantities that vary smoothly

A continuous random variable takes on an uncountably infinite number of possible values. For a discrete random variable 92X92 that takes on a finite or countably infinite number of possible values, we determined 92PXx92 for all of the possible values of 92X92, and called it the probability mass function quotp.m.f.quot.

Probability Distributions for Continuous Variables Definition Let X be a continuous r.v. Then a probability distribution or probability density function pdf of X is a function f x such that for any two numbers a and b with a b, we have The probability that X is in the interval a, b can be calculated by integrating the pdf of the r.v. X.

The continuous random variable formulas for these functions are given below. PDF of Continuous Random Variable. The probability density function of a continuous random variable can be defined as a function that gives the probability that the value of the random variable will fall between a range of values. Let X be the continuous random

Definition density function. The probability distribution of a continuous random variable 92X92 is an assignment of probabilities to intervals of decimal numbers using a function 92fx92, called a density function, in the following way the probability that 92X92 assumes a value in the interval 9292left a,b92right 92 is equal to the area of the region that is bounded above by the graph of

Continuous. Random Variables can be either Discrete or Continuous Discrete Data can only take certain values such as 1,2,3,4,5 It has equal probability for all values of the Random variable between a and b The probability of any value between a and b is p. We also know that p 1b-a, because the total of all probabilities must be 1

I For a continuous random variable, PX x 0, the reason for that will become clear shortly. I For a continuous random variable, we are interested in probabilities of intervals, such as Pa X bwhere a and b are real numbers. I We will introduce the probability density function pdf to calculate probabilities, such as Pa X b 357

LECTURE 8 Continuous random variables and probability density functions Probability density functions . Properties . Examples Expectation and its properties The expected value rule Linearity Variance and its properties Uniform and exponential random variables Cumulative distribution functions Normal random variables

Definition Continuous Random Variable. Definition We say that a random variable 92X92 has a continuous distribution or that 92X92 is a continuous random variable if there exists a nonnegative function 92f92, defined on the real line, such that for every interval of real numbers, the probability that 92X92 takes a value in the interval is the integral of 92f92 over the interval.