Question

This problem set concerns the pdf fX​(x;a,b)={C(a,b)(b−x)(x−a)0​ if a

Answer 1

The probability density function
`(pdf) \(f_X(x;a,b)\)` is given by
`\(f_X(x;a,b) = C(a,b)(b-x)(x-a)\)` if
`\(a < x < b\)` and
`\(0\)` otherwise.

Step-by-step explanation:

The given probability density function
`\(f_X(x;a,b)\)` represents a continuous probability distribution over the interval
`\((a, b)\)`. To understand this, let's break down the components of the function.

The expression
`\((b-x)(x-a)\)` suggests a parabolic shape, indicating that the probability is higher towards the middle of the interval and decreases towards the edges.

The constants
`\(a\)` and
`\(b\)` define the boundaries of the interval, with
`\(a < x < b\)`. The multiplication by
`\(C(a,b)\)`ensures that the function integrates to 1 over the specified interval, making it a valid probability density function.

The function is
`\(0\)` outside the interval , implying that the random variable
`\(X\)` can only take values within this range.

The parabolic shape ensures that values closer to the center of the interval are more probable, creating a symmetric distribution. The role of
`\(C(a,b)\)` is crucial as it normalizes the function, making the total area under the curve equal to 1.

In summary, the given probability density function
`\(f_X(x;a,b)\)` describes a continuous probability distribution over the interval
`\((a, b)\)` with a parabolic shape, emphasizing higher probabilities in the middle of the interval.

The constants
`\(a\)` and
`\(b\)` define the range of possible values for the random variable
`\(X\)`, and
`\(C(a,b)\)` ensures the proper normalization for a valid probability density function.