A probability distribution for the random variable X, the...

A probability distribution for the random variable $X$, the number of trials until a success is observed, is called the geometric probability distribution. It has the same criteria as the binomial distribution (see page 334 ), except that the number of trials is not fixed. Its probability distribution function (pdf) is $$P(x)=p(1-p)^{x-1}, \quad x=1,2,3, \ldots$$ where $p$ is the probability of success. (a) What is the probability that Shaquille O'Neal misses his first two free throws and makes the third? Over his career, he made $52.4 \%$ of his free throws. That is, find $P(3)$. (b) Construct a probability distribution for the random variable $X,$ the number of free-throw attempts of Shaquille O'Neal until he makes a free throw. Construct the distribution for $x=1,2,3, \ldots, 10 .$ The probabilities are small for $x>10.$ (c) Compute the mean of the distribution, using the formula presented in Section 6.1. (d) Compare the mean obtained in part (c) with the value $\frac{1}{p} .$ Conclude that the mean of a geometric probability distribution is $\mu_{X}=\frac{1}{p} .$ How many free throws do we expect Shaq to take before we observe a made free throw?

A probability distribution for the random variable $X$, the number of trials until a success is observed, is called the geometric probability distribution. It has the same criteria as the binomial distribution (see page 334 ), except that the number of trials is not fixed. Its probability distribution function (pdf) is
$$P(x)=p(1-p)^{x-1}, \quad x=1,2,3, \ldots$$
where $p$ is the probability of success.
(a) What is the probability that Shaquille O'Neal misses his first two free throws and makes the third? Over his career, he made $52.4 \%$ of his free throws. That is, find $P(3)$.
(b) Construct a probability distribution for the random variable $X,$ the number of free-throw attempts of Shaquille O'Neal until he makes a free throw. Construct the distribution for $x=1,2,3, \ldots, 10 .$ The probabilities are small for $x>10.$
(c) Compute the mean of the distribution, using the formula presented in Section 6.1.
(d) Compare the mean obtained in part (c) with the value $\frac{1}{p} .$ Conclude that the mean of a geometric probability distribution is $\mu_{X}=\frac{1}{p} .$ How many free throws do we expect Shaq to take before we observe a made free throw?

Key Concepts

Geometric Distribution

A discrete probability distribution that models the number of independent Bernoulli trials conducted until the first success is observed. It is characterized by its probability mass function, P(X = x) = p(1-p)^(x-1), where x is a positive integer and p is the success probability on each trial.

Probability Mass Function (PMF)

A function that assigns the probability that a discrete random variable is exactly equal to a specific value. In the context of the geometric distribution, the PMF determines the likelihood of the first success occurring on the x-th trial.

Expected Value (Mean) of a Distribution

A measure of the central tendency or average outcome of a random variable. For the geometric distribution, the expected value is calculated as the sum of x multiplied by its probability, yielding a mean of 1/p, which represents the average number of trials needed to achieve a success.

Bernoulli Trials

A sequence of independent experiments in which each experiment (trial) results in a binary outcome (success or failure) with a constant probability of success. The geometric distribution arises naturally from sequences of Bernoulli trials, where one is interested in the number of trials required to obtain the first success.

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