For the cumulative distribution function \(F(x) = \left\{ {\begin{array}{*{20}{c}} {0;x < - 1}\\ {\frac{1}{2}{{(x + 1)}^2}; - 1 \le x < 0}\\ {1 - \frac{{{{(1 - x)}^2}}}{2};0 \le x < 1}\\ {1.1 < x < \infty } \end{array}} \right.\)

the upper quartile point is

The correct answer is 1-\(\sqrt{0.5}\)

Key Points

• The cumulative distribution function (CDF) is a function that describes the probability distribution of a random variable.
  • In this case, the CDF is defined by the function F(x), which has different values for different ranges of x.
• The upper quartile point of a data set is the value that corresponds to the 75th percentile.
• To calculate the upper quartile point using the CDF, we need to find the value of x for which F(x) is equal to 0.75.
  • To understand this, we can look at the formula for F(x) for the range \(0 ≤ x < 1\).
  • We can see that this part of the function is a quadratic function with a maximum value of 1 at x = 0.
  • In this case, we can see that F(x) is equal to 0.75 when x is equal to\( 1 - √(1/2)\).

Additional Information

• The upper quartile corresponds to the value of x at which F(x) is equal to 0.75, or the point where 75% of the data is below this value.
• Since the maximum value of the quadratic function is at x = 0, we can calculate the upper quartile by finding the root of the quadratic equation\( 1 - (1-x)^2/2 = 0.75\), which simplifies to\( (1-x)^2 = 0.5\).
• Solving for x, we get x = 1 - √(1/2), which is the upper quartile point.
• Overall, the CDF can be a useful tool for analyzing and modeling probability distributions, and the upper quartile point is a useful measure of variability that can provide insight into the spread and shape of a data set.