Question

Determine the constant \(c\) so that \(f(x)=c x(3-x)^{4}, 0

Short answer

The value of the constant that makes \(f(x)=cx(3-x)^{4}\) a pdf is \(c = \frac{1}{405}\).

Step-by-step solution

Write down the integral equation

To ensure \(f(x)\) is a pdf, the integral of \(f(x)\) over its entire domain must equal 1. The appropriate set-up is: \[ \int_{-\infty}^{+\infty} f(x)dx = \int_{0}^{3} c x(3-x)^{4} dx = 1 \]

Solve the integral

Now the integral has to be solved. This is an application of the power rule for integration along with a simple u-substitution. Let's let \(u = 3 - x\), so \(du = -dx\). The integral becomes: \[ \int_{0}^{3} c x(3-x)^{4} dx = -c\int_{3}^{0} u^4(3-u) du \] Simplifying this, we get: \[ -c[243 - 405u + 270u^2 - 90u^3 + 9u^4]_{0}^{3} = -c[0 - 0 + 0 - 0 + 0- (243 - 405*3 + 270*3^2 - 90*3^3 + 9*3^4)] = -c*(-405) = 405c \]

Equate the integral to 1

Now, in order to satisfy the condition that the integral of the pdf equals 1, equate 405c and 1, such that \[405c = 1 \]. Solving this equation gives the value of the constant \(c\).

Solve for the constant \(c\)

Solving the equation \(405c = 1\) for \(c\) gives \[c = \frac{1}{405}\]

Key concepts

Integral Calculus

Integral calculus is a branch of mathematics that involves finding the integral of functions. It is concerned with the accumulation of quantities, such as areas under curves, volumes of solids, and accumulated change. In the context of probability, the integral calculus is used to find the total probability distributed across a continuous random variable which is represented by a probability density function (pdf).

For a function to qualify as a pdf, the total area under its curve between the limits of the random variable must equal 1, reflecting the fact that the total probability of all outcomes must sum up to 1. This means we integrate the function across its bounds, and if this integral equals 1, the function is a legitimate pdf. In the given exercise, to determine the constant c for our function to be a pdf, we verify that the integral from 0 to 3 of the function f(x) equals 1.

U-Substitution in Integration

The u-substitution is a technique used in integral calculus to simplify the integration of composite functions. It's akin to the reverse process of the chain rule in differentiation. Essentially, we are choosing a substitution, say u, to replace a part of the integrand, breaking down a complex problem into a simpler one.

In our exercise, to find the constant c for f(x), we used u-substitution by letting u equal (3 - x). This substitution transforms the integral into a new form involving just u, making it easier to handle since it now only contains powers of u. The limits of integration also change according to the new variable u. Then we can integrate more conveniently using basic rules of integration.

Power Rule for Integration

The power rule for integration is one of the fundamental rules that enable us to integrate any term in the form of u^n where n is a real number not equal to -1. The power rule states that the integral of u^n with respect to u is (u^(n+1))/(n+1), plus a constant of integration in indefinite integrals.

In the step-by-step solution provided, after we make the u-substitution, we apply the power rule to integrate terms such as u^4 and (3-u). This step significantly simplifies the process compared to attempting to directly integrate the original function. Once integrated, we evaluate the definite integral by substituting the limits of integration. The resulting expression involving c allows us to solve for the value of the constant that makes the integral of the proposed function equal to 1, thus confirming it as a pdf.