Find the value of c that makes the function a probability density function

TL;DR

Calculate the value of c to make a piecewise function a probability density function.

Transcript

in this problem we have a function f of y it's a piecewise function and we have to find the value of c that makes it a probability density function so solution so there's two properties that have to be satisfied the first one is that f of y is non-negative for all values of y and the second one is if we integrate from negative infinity to infinity ... Read More

Key Insights

• 🚱 Probability density functions must be non-negative for all values.
• 🟰 The integral of a probability density function over its domain should equal one.
• 🆘 Splitting the integral of a piecewise function helps in finding the value of c.
• 🆘 Evaluating the integral over specific intervals helps in determining the value that satisfies the conditions.
• 😫 Understanding how to set up and solve for c in a probability density function is crucial in statistics.
• ✅ Checking the function with the calculated value of c ensures it meets the required conditions.
• 😀 By integrating the piecewise function correctly, the value of c can be accurately determined.

Questions & Answers

Q: What are the two properties that must be satisfied for a function to be a probability density function?

The function must be non-negative for all values and its integral from negative infinity to infinity must equal one.

Q: How is the integral of a piecewise function used to find the value of c?

By setting the integral to one and evaluating the integral over the intervals where the function is non-zero, we can solve for the value of c.

Q: Why do we split the integral into three parts when finding the value of c?

By evaluating each segment separately, we can ensure that the function satisfies the properties of a probability density function over its defined intervals.

Q: How does verifying the value of c after integration confirm it as a probability density function?

Substituting the calculated value of c back into the function and checking if it is non-negative for the given intervals ensures it meets the criteria for a probability density function.

Summary & Key Takeaways

• Function f(y) must be non-negative for all y values.

• Integral of f(y) from negative infinity to infinity should equal one.

• Solving for c, integrate the piecewise function to find c = 1/2.