Density of continuous random variables?

In summary, the density of a continuous random variable is a function that represents the likelihood of the variable taking on a specific value within a given interval. It is different from the probability distribution function (PDF) as it is the derivative of the cumulative distribution function (CDF). The area under the curve of the density function represents the probability of the variable falling within a specific interval, with a total area of 1. The density is calculated by taking the derivative of the CDF, which is obtained by integrating the PDF. It is always non-negative, as probabilities cannot be negative.
  • #1
icup007
2
0
Can you please help me find the density of the following functions?

The density of an absolutely continuous random variable X is:

fX(x) =
{ (3x^2-1)/12 if 1<x<2
{ 1/2 if 2<x<3
{ 0 elsewhere

Find the density of Y where Y = 4X-2
Find the density of M where M = (X-2)^2


Thank you!
 
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  • #2

FAQ: Density of continuous random variables?

1. What is the definition of density for a continuous random variable?

The density of a continuous random variable is a function that assigns probabilities to the different values that the variable can take on. It represents the likelihood of the variable taking on a specific value within a given interval.

2. How is the density function different from the probability distribution function (PDF)?

The density function is the derivative of the cumulative distribution function (CDF), which is the integral of the probability distribution function. The PDF is used to calculate the probability of a continuous random variable falling within a specific range of values, while the density function is used to calculate the probability of a single value occurring.

3. What is the relationship between the density function and the area under the curve?

The density function is represented by a curve on a graph, and the area under this curve represents the probability of the continuous random variable falling within a specific interval. The total area under the curve is equal to 1, as the probability of the variable taking on any value must be 1.

4. How is the density of a continuous random variable calculated?

The density of a continuous random variable is calculated by taking the derivative of the cumulative distribution function (CDF). The CDF is calculated by integrating the probability distribution function (PDF) over a specific range of values. The resulting derivative gives the value of the density function at a given point.

5. Can the density function of a continuous random variable take on negative values?

No, the density function of a continuous random variable must always be non-negative. This is because the density function represents the probability of the variable taking on a specific value, and probabilities cannot be negative. If the density function were to take on negative values, it would violate this fundamental principle of probability theory.

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