Question

Assume the pdf of a continuous random variable X is:

(x) = {
0 x < −1
0.2 − 1 ≤ x < 0
0.8 0 ≤ x < 1
0 x ≥ 1
}

a. Write the equation that defines the associated cumulative distribution function (cdf).
b. Compute the expected value, E[X]
c. Compute the variance, Var[X]
d. Compute the coefficient of variation, δX
e. Compute the skewness

Answer 1

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.