Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.
f(x) = (1/x) over [c, c + 1]

검증된 단계별 안내

Recall that for a function \(f(x)\) to be a probability density function (pdf) over an interval \([a, b]\), it must satisfy the condition: \(\int_a^b f(x) \, dx = 1\).

In this problem, the function is \(f(x) = \frac{1}{x}\) defined over the interval \([c, c+1]\). We need to find the value of \(c\) such that \(\int_c^{c+1} \frac{1}{x} \, dx = 1\).

Set up the integral: \(\int_c^{c+1} \frac{1}{x} \, dx = \left[ \ln|x| \right]_c^{c+1} = \ln(c+1) - \ln(c)\).

Use the logarithm property to combine the difference: \(\ln(c+1) - \ln(c) = \ln\left( \frac{c+1}{c} \right)\).

Set the integral equal to 1 and solve for \(c\): \(\ln\left( \frac{c+1}{c} \right) = 1\). From here, exponentiate both sides to isolate \(c\) and solve the resulting equation.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Probability Density Function (PDF)

A probability density function describes the likelihood of a continuous random variable taking on a specific value. For a function to be a valid PDF, it must be non-negative over its domain and its total integral over the specified interval must equal 1.

Definite Integral and Area Under the Curve

The definite integral of a function over an interval represents the area under its curve between two points. In the context of PDFs, integrating the function over its domain must yield 1, ensuring the total probability sums to one.

Solving for Constants Using Integral Equations

When a function includes an unknown constant, we use integral equations to solve for it by setting the integral equal to 1 (for PDFs). This involves integrating the function with the constant over the interval and solving the resulting equation.

Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.f(x) = (1/x) over [c, c + 1]