Find E[X] and Var[X] for a Pascal random variable X with...

Find $E[X]$ and $\operatorname{Var}[X]$ for a Pascal random variable $X$ with parameters $r$ and $p$. Hint: Use Theorem 2.9.2(c) and the fact that, by the result of Exercise 35(d) of Chapter 2, Formula (3.211) can be written as $$ p(k ; r, p)=\left(\begin{array}{c} -r \\ k \end{array}\right) p^r(-q)^r \quad k=0,1, \ldots $$

Find $E[X]$ and $\operatorname{Var}[X]$ for a Pascal random variable $X$ with parameters $r$ and $p$. Hint: Use Theorem 2.9.2(c) and the fact that, by the result of Exercise 35(d) of Chapter 2, Formula (3.211) can be written as
$$
p(k ; r, p)=\left(\begin{array}{c}
-r \\
k
\end{array}\right) p^r(-q)^r \quad k=0,1, \ldots
$$

Key Concepts

Probability Generating Functions

Probability generating functions (PGFs) are used as a powerful tool in the analysis of discrete random variables. They are particularly useful for calculating moments such as the expectation and variance by differentiating the PGF. In the case of the Pascal distribution, the generating function allows for an elegant derivation of these moments by linking the coefficients of the series expansion of the PGF with the probabilities in the distribution.

Variance

Variance measures the spread or dispersion of a random variable's distribution around its mean. In the context of the Pascal distribution, the variance quantifies how much the number of failures before achieving r successes is expected to fluctuate. It is calculated by finding the expected value of the square of the random variable and then subtracting the square of the expectation. Techniques involving generating functions and differentiation can simplify this computation.

Expectation

Expectation, or the mean of a random variable, is a measure of the central tendency of its probability distribution. For the Pascal distribution, the expectation represents the average number of failures observed before reaching the rth success. It is computed using the definition of expected value, typically through the summation over all possible outcomes weighted by their corresponding probabilities or more efficiently via moment generating functions or differentiation of the probability generating function.

Pascal Distribution

The Pascal distribution, also known as the negative binomial distribution, describes the number of failures that occur in a sequence of independent Bernoulli trials before a specified number of successes is reached. It is a discrete probability distribution that is parameterized by the number of successes (r) and the success probability (p) in each trial. This distribution finds applications in various fields where processes continue until a predetermined number of successful outcomes have been observed.

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