Question

EXAMPLE 2.4 If an individual is selected at random from a large group of adult males, the probability that his height X is precisely 68 inches (i.e., 68.000… inches) would be zero. However, there is a probability greater than zero than X is between 67.000… inches and 68.500… inches, for example. A function f(x) that satisfies the above requirements is called a probability function or probability distribution for a continuous random variable, but it is more often called a probability density function or simply density function. Any function f(x) satisfying Properties 1 and 2 above will automatically be a density function, and required probabilities can then be obtained from (8).

Answer 1

The probability density function (pdf) is used in probability theory for continuous random variables. The area under the pdf curve represents probabilities, with the entire area equaling one. Probability for a continuous variable is expressed as the chance of falling within a range rather than at a specific point.

Step-by-step explanation:

Continuous Probability Functions

The psychological concept described is the probability density function (pdf), which is crucial in understanding probabilities associated with continuous random variables.

Unlike a discrete random variable that takes on countable values, a continuous random variable takes on an uncountable infinity of possible values within an interval, which are obtained through measurements.

Due to these infinite possibilities, the probability of a continuous variable having a precise value is zero; instead, we focus on the probability of a variable falling within a range of values.

The pdf f(x) is such that the area under the curve between any two points (a and b) on the x-axis represents the probability that the variable falls between those values, P(a ≤ x ≤ b).

The total area under the probability density function is always one, reflecting the fact that the sum of all probabilities in a distribution equals one.

Additionally, the cumulative distribution function (cdf), which provides the probability that a random variable is less than or equal to a certain value, is related to the pdf but focuses on cumulative probability rather than density at a specific point.