Question

A parabolic satellite dish reflects signals to the dish's focal point. An antenna designer analyzed signals transmitted to a satellite dish and obtained the probability density function f(x)=c(1− 1/16x²) for 0a. Value of c that makes f(x) a valid probability density function.

b. the cumulative distribution function, F(x).
c. P(0.1d. Expected Value, E(X) and Variance, V(X).

Answer 1

The student's question requires finding the value of c for the given probability density function, as well as the cumulative distribution function, a specific probability, and the expected value and variance of the distribution.

Step-by-step explanation:

To solve the student's problem regarding the probability density function (pdf), cumulative distribution function (cdf), and related probabilities and statistics, we first need to ascertain the value of c that makes f(x) a valid pdf. Since the total area under a pdf must equal 1, we set up an integral from 0 to 4 of c(1 - 1/16x²) dx equal to 1 and solve for c.

Next, we determine the cumulative distribution function, F(x), by integrating the pdf from 0 to x. This will give us the probability that X is less than or equal to a certain value x.

To find P(0 < x < 4), we would integrate the pdf over the interval [0, 4], which should result in 1 based on the properties of pdfs.

Finally, the Expected Value, E(X), is calculated by integrating x multiplied by the pdf over the range of X, and the Variance, V(X), is calculated using E(X²) - [E(X)]², where E(X²) is the expected value of X squared.