Calculo de probabilidades usando la f.d.p. de una V.A. continua, ejemplo 3

Continuing with Probability Functions

Understanding Probability Density Functions

• The probability that a variable is less than or equal to a certain number can be calculated using the integral of the probability density function from negative infinity to that number. This involves finding the area under the curve up to that point.

• For this example, we need to calculate the integral from 0 to 3 instead of negative infinity, as integrating from zero simplifies our calculations.

• The probability density function changes between intervals: from 0 to 2 it is 1/4x , and from 2 to 4 it becomes 1 - 1/4x . Thus, we must split our integration into two parts.

• We will compute two integrals: one for x ranging from 0 to 2 and another for x ranging from 2 to 3. The first integral evaluates 1/4x^2/2 , while the second involves integrating (1 - 1/4x)dx .

Evaluating Integrals

• The evaluation of these integrals leads us through specific calculations where constants are maintained throughout. For instance, evaluating at limits gives us results based on squared terms divided by constants.

• After performing evaluations at specified limits, we simplify expressions involving fractions and whole numbers, leading us toward calculating probabilities effectively.

• Ultimately, after simplification, we arrive at a fraction representing the probability of being less than or equal to three.

Alternative Methods for Calculation

• An alternative method involves using complementary probabilities. To find the probability that x leq 3, we can subtract the probability of x > 3 from one.

• By recognizing that beyond a certain point (in this case above three), the function's value is zero simplifies our calculations significantly since only one integral needs evaluation in this scenario.

• This approach allows us to focus on calculating just one integral over an interval where the density function remains constant rather than dealing with multiple segments.

Final Calculations

• As we integrate over defined limits (from three onwards), we apply known values and evaluate them against established functions within those bounds.

Mathematical Operations and Simplifications

Understanding the Expression

• The speaker begins by discussing a mathematical expression, indicating that it simplifies to 1.

• They mention performing operations involving subtraction: "cuatro menos dos dados menos 30" (four minus two given minus thirty).

• The result of these calculations leads to an intermediate value of -1, which is then adjusted with fractions: "menos uno más de los octavos" (minus one plus eighths).

• The speaker explains the need for further adjustments in the expression, suggesting that additional terms will be added or subtracted.