Final answer:
The student's question involves calculating various attributes of a continuous random variable's distribution, including its cumulative distribution function, expected value, variance, coefficient of variation, and skewness. The calculations are based on the given probability density function (pdf).
Step-by-step explanation:
The student's question concerns the construction and evaluation of a probability density function (pdf), cumulative distribution function (cdf), expected value (mean), variance, coefficient of variation, and skewness of a continuous random variable X with a specified pdf.
The pdf of X provided is:
- 0 for x < -1
- 0.2 for -1 ≤ x < 0
- 0.8 for 0 ≤ x < 1
- 0 for x ≥ 1
(a) The cumulative distribution function (cdf) for a continuous random variable is defined as the probability that X is less than or equal to a certain value, expressed as a function of x:
F(x) = ∫_{-∞}^{x} f(t) dt. To find the cdf, we integrate the pdf over the specified intervals.
(b) The expected value of X, E[X], is the probability-weighted average of all possible values of X. It can be calculated as the integral of x multiplied by the pdf over the entire range of X.
(c) The variance of X, Var[X], is a measure of the spread of the distribution and is calculated as the integral of the squared deviation from the mean (x - E[X])^2 times the pdf over the entire range of X.
(d) The coefficient of variation, δX, is the ratio of the standard deviation to the mean, providing a normalized measure of dispersion relative to the mean.
(e) Skewness is a measure of the asymmetry of the probability distribution of X. It can be calculated using the third central moment.