Final answer:
The values for a, b, and c that make each pdf valid are 2/9, 3/8, and 4 respectively. The expected values for the random variables X, Y, and Z are 2, 1.5, and 0.8 respectively. The cumulative distribution functions are the integrated areas of the pdfs up to a variable upper limit.
Step-by-step explanation:
Continuous Probability Functions
To determine the appropriate values for a, b, and c that make each pdf a valid probability density function, we need to ensure that the total area under each probability density function (pdf) is equal to 1. This is because the total probability must sum to 1 for any probability distribution.
Finding Values for a, b, and c
For X, we integrate the given pdf over the interval from 0 to 3:
\(\int_{0}^{3} ax \ dx = 1\)
Solving this, we find that a = 1/ \tfrac{3^2}{2} or a = 2/9.
For Y, we integrate the given pdf from 0 to 2:
\(\int_{0}^{2} bx^2 \ dx = 1\)
Solving this, we find that b = 1/ \tfrac{2^3}{3} or b = 3/8.
For Z, we integrate the pdf from 0 to 1:
\(\int_{0}^{1} cx^3 \ dx = 1\)
Solving this, we find that c = 1/ \tfrac{1^4}{4} or c = 4.
Calculating Expected Values
The expected value (mean) for each variable is calculated by integrating the product of the variable and its pdf over the possible values.
For X:
\(\mu_X = \int_{0}^{3} x(ax) \ dx\)
After substituting a and integrating, \(\mu_X = 2\).
For Y:
\(\mu_Y = \int_{0}^{2} x(bx^2) \ dx\)
After substituting b and integrating, \(\mu_Y = 1.5\).
For Z:
\(\mu_Z = \int_{0}^{1} x(cx^3) \ dx\)
After substituting c and integrating, \(\mu_Z = 0.8\).
Constructing Cumulative Distribution Functions
The cumulative distribution function (cdf) for each random variable is obtained by integrating the pdf up to a variable upper limit.
Sketching PDF and CDF
The graph of the pdf and cdf can be sketched by plotting the pdf and then drawing the corresponding cdf as the integrated area under the pdf curve.