# Probability density functions

Defining Probability Density Functions

• The function which describes a continuous random variable is called a Probability Density Function (pdf).
• Animated by the laws of probability, the area under the pdf must always equal 1.
• PDF cannot take negative values, as negative probability doesn’t exist in natural logic.

Understanding Probability Density Functions

• A point on the pdf does not represent the probability of that outcome in itself, but the area under the curve in an interval represents the probability of the outcome falling within that interval.
• Specific outcomes have a zero probability in continuous random variables because there are infinite possibilities within the range.
• Simply put, when considering a range of values, the probability of the variable falling within that range can be determined by measuring the area under the curve of the pdf within that range.

Working with Probability Density Functions

• Determining a range of values can be done graphically. Read the corresponding values on the x-axis and find the area that falls within this range on the pdf.
• Alternatively, calculating the area requires integration in calculus, taking the integral of the pdf over the desired interval.
• The integration can be performed from x=a to x=b to get the probability of the event lying in this range.
• Note that a basic understanding of integral calculus is necessary for this aspect of continuous random variables, so revisiting previous maths work may be beneficial.

Integral of Probability Density Functions

• An integral of a pdf from a specific value to infinity is equal to the probability that this continuous random variable takes a value greater than or equal to that specific value.
• Its analogue, an integral from negative infinity to the specific value equals the probability that the random variable takes a value smaller or equal to that value.
• These calculations are important for understanding other key concepts such as cumulative density functions and expected values.

Remember to apply both mathematical knowledge and logic when dealing with probability density functions. Understanding the graph and the calculations can be key for successful completion of advanced mathematics problems.