Let X be a random variable of the discrete type with pmf...

Let $X$ be a random variable of the discrete type with pmf $p(x)$ that is positive on the nonnegative integers and is equal to zero elsewhere. Show that
$$
E(X)=\sum_{x=0}^{\infty}[1-F(x)]
$$
where $F(x)$ is the cdf of $X$.

Key Concepts

Discrete Random Variable

A discrete random variable takes on countable values, typically integers. It is characterized by a probability mass function that assigns a probability to each possible value, and its behavior is analyzed using sums rather than integrals.

Probability Mass Function (pmf)

The probability mass function specifies the probability that the random variable equals a particular value. It is fundamental in computing probabilities and expectations, as it provides the weight of each outcome in the sample space.

Cumulative Distribution Function (cdf)

The cumulative distribution function gives the probability that the random variable takes a value less than or equal to a particular number. It accumulates the probabilities specified by the pmf and is used to describe the distribution of a random variable in a cumulative manner.

Expectation

The expectation, or mean, of a random variable is a measure of its central tendency. For a discrete random variable, it is computed as the sum of each possible value weighted by its probability. It is a key concept in understanding the long-run average outcome of a random process.

Tail Sum Formula

The tail sum formula expresses the expectation of a nonnegative discrete random variable as the sum of the tail probabilities, that is, the probabilities that the random variable exceeds or equals a certain threshold. This alternative formulation is particularly useful in situations where computing the sum of tail probabilities is more straightforward than summing over the weighted outcomes.

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