Question

Let \(X\) be a random variable of the continuous type with pdf \(f(x)\), which is positive provided \(0

Short answer

The expectation of the random variable \(X\) from \(0\) to \(b\) can be expressed in terms of the cumulative distribution function \(F(x)\) as \(E(X)=\int_{0}^{b}[1-F(x)] dx\).

Step-by-step solution

Write down the definition of expectation

The expectation of a random variable \(X\), \(E(X)\), is defined as the integral of \(x\) times \(f(x)\) where \(f(x)\) is the probability density function of \(X\). The range of integration is from 0 to \(b\). So, \(E(X) = \int_{0}^{b}x f(x) dx\).

Apply the integration by parts

The method of integration by parts states that the integral of the product of two functions is equal to the product of the first function and the integral of the second function minus the integral of the derivative of the first function times the integral of the second function. Let's apply it to the expression for \(E(X)\). Choose \(u = x\) and \(dv = f(x) dx\). Then, \(du = dx\) and \(v = \int f(x) dx = F(x)\). According to the rule of integration by parts, \(\int u dv = uv - \int v du\), we get \(E(X) = [xF(x)]_{0}^{b} - \int_{0}^{b}F(x) dx\).

Simplify the equation

We simplify the above equation. The first term equals \([bF(b) - 0F(0)]\). Now, it is given that \(b\) is the maximum value \(X\) can take and at this value the cumulative distribution function \(F(b)\) is 1. Also, \(F(0)\) is zero because the cumulative distribution function at the minimum value is always zero. The first term thus simplifies to \(b - 0 = b\). In the second term, we can separate \(1-F(x)\) as its own integral minus the integral of \(F(x)\): \(- \int_{0}^{b}F(x) dx = - \int_{0}^{b}dx + \int_{0}^{b}[1-F(x)] dx\). The first integral on the right hand side equals \(b\). Hence, \(E(X) = b - b + \int_{0}^{b}[1-F(x)] dx = \int_{0}^{b}[1-F(x)] dx\). Thus, the proof is complete.

Key concepts

Probability Density Function (pdf)

A Probability Density Function (pdf) is a core concept in statistics that describes the likelihood of a continuous random variable taking on a particular value. Think of it like this: If you have a smooth curve that stretches along an axis representing all possible values the random variable can take, the area under any part of that curve represents the probability of finding the variable within that range.

For a continuous random variable, the pdf is crucial because it helps us calculate probabilities for intervals rather than for discrete points. The pdf is denoted as f(x), and the probability that a variable falls between two points, a and b, is found by integrating the pdf over that interval:
\[ P(a < X < b) = \int_a^b f(x) dx. \]
However, it is essential to remember that the pdf must satisfy two conditions: it is non-negative over its entire range, and the total area under the curve is equal to 1, ensuring that it represents a valid probability distribution.

Cumulative Distribution Function (cdf)

The Cumulative Distribution Function (cdf) is another significant concept tied closely to the pdf. It accumulates the probabilities up to a certain value, painting a picture of the overall distribution. Essentially, the cdf, denoted by F(x), tells us the probability that a random variable X is less than or equal to a specific value x.

Mathematically, the cdf is defined as:\[ F(x) = P(X \leq x) = \int_{-\infty}^{x} f(t) dt. \]
It starts at 0 and monotonically increases to 1 as x approaches infinity. The cdf is useful because it's often easier to work with when calculating probabilities for intervals and is the foundation for other statistical concepts, like percentiles and the median. Moreover, understanding the cdf is critical when proving certain properties of random variables, as it’s integral to the calculation of expectations and variances.

Integration by Parts

Integration by Parts is a powerful technique in calculus, often used when dealing with the product of two functions that are not easily integrable. It's like the product rule for differentiation, but in reverse.

The formula for integration by parts can be remembered by the acronym ILATE (Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential), which helps in choosing u and dv wisely:\[ \int u dv = uv - \int v du. \]
When faced with an expectation that requires the multiplication of x (an algebraic function) with the pdf f(x), this method breaks down the problem into more manageable parts. By carefully selecting u and dv, you can convert the integration of a product of functions into simpler terms, as demonstrated in the exercise. It's a fundamental technique that not only streamlines complex integrations but also connects various parts of the probability theory, enhancing our understanding of random variables.