Question

F X ​ (x)= ⎩ ⎨ ⎧ ​ 0 2 1 ​ [1+sin( 4 π ​ x)] 1 ​ −[infinity]

Answer 1

The function
`\( F_X(x) \)` is a piecewise-defined distribution function. It equals 0 for
`\( x < 0 \)`,
`\((1)/(2) + (1)/(2)\sin(4\pi x)\) for \(0 \leq x \leq 1\)`, and
`1 for \(x > 1\)`.

Step-by-step explanation:

The given function
`\( F_X(x) \)` is defined in three different intervals. For
`\(x < 0\)`, the function is 0, representing that the cumulative distribution function (CDF) is 0 before the lower bound of the variable. In the interval
`\(0 \leq x \leq 1\)`, the function is
`\((1)/(2) + (1)/(2)\sin(4\pi x)\)`, indicating a sinusoidal increase from 0.5 to 1, creating a non-linear cumulative distribution. This means that as
`\(x\)` increases in this interval, the probability accumulates in a sinusoidal manner. For
`\(x > 1\)`, the function is 1, denoting that the CDF reaches its maximum value of 1 after the upper bound of the variable. This step function behavior is typical for cumulative distribution functions, where the value increases in steps at the points where the variable is defined.

In summary,
`\( F_X(x) \)` is a non-standard distribution with a sinusoidal increase in probability between 0 and 1, demonstrating a unique pattern in how the cumulative probability accumulates over the range of the variable. This type of distribution function is useful in modeling scenarios where the probability of an event varies in a non-linear fashion across different intervals of the variable.