Variable aleatoria continua - Hallar c para que sea f sea función de densidad de probabilidad f.d.p.

Understanding Probability Density Functions

Introduction to Probability Density Functions

• The video discusses how to determine a parameter in a function so that it qualifies as a probability density function (PDF) .

• A PDF must be positive and, for continuous random variables, the integral from negative infinity to infinity of the density function must equal 1 .

Setting Up the Integral

• The limits of integration depend on where the density function is non-zero; in this case, it's between 0 and 4 .

• The integral can be split into two parts: one from 0 to 2 and another from 2 to 4, reflecting different expressions for the PDF in these intervals .

Evaluating Integrals

• For x between 0 and 2, the expression involves integrating ax^2/2 , while for x between 2 and 4, it changes to c(4 - x) .

• Calculating these integrals will help find constants needed for normalization. The first integral evaluates straightforwardly while the second requires substituting values at specific points .

Solving for Constants

• After evaluating both integrals, calculations yield results that need simplification. This includes subtracting terms derived from evaluating at upper and lower limits of integration .

• Ultimately, solving leads to determining that c = 1/4 , ensuring that the total area under the PDF equals one .

Finalizing the Probability Density Function

• The final form of the PDF is established as:

• f(x) = 1/4x for 0 < x < 2

• f(x) = 1/4(4 - x) for 2 < x < 4

• Zero elsewhere .