Problem 97 - Essential Calculus

Question

Probability In Exercises 97 and 98, the random variable \(\boldsymbol{n}\) represents the number of units of a product sold per day in a store. The probability distribution of \(n\) is given by \(P(n) .\) Find the probability that two units are sold in a given day \([P(2)]\) and show that \(P(1)+P(2)+P(3)+\cdots=1\). $$ P(n)=\frac{1}{2}\left(\frac{1}{2}\right)^{n} $$

Step-by-Step Solution

Answer

The probability that two units are sold in a given day is \(P(2) = 1/8\). The sum of all probabilities equals to 1 which confirms the fact that the given function is indeed a probability distribution.

Step 1: Evaluate the Probability of Selling 2 units

We have to find out the probability of selling 2 units per day. Here, the random variable \(n\) is equal to 2. We can find this probability by substituting \(n=2\) into the given probability function which is \(P(n) = \frac{1}{2}\left(\frac{1}{2}\right)^{n}\) : \[ P(2) = \frac{1}{2}\left(\frac{1}{2}\right)^{2}.\] Multiply the terms together to get the final probability.

Step 2: Prove the Summation of All Probabilities Equals 1

According to the properties of probabilities, the sum of probabilities of all possible outcomes should add up to 1. This means that \(P(1)+P(2)+P(3)+\cdots=1.\) The given \(P(n)\) is a geometric series with common ratio of \(1/2\). Using the sum formula for geometric series \(S = \frac{a}{1-r}\), where \(a\) represents the first term and \(r\) is the common ratio, we will substitute the respective values \(a = P(1) = \frac{1}{2}\left(\frac{1}{2}\right)^{1}\) and common ratio \(r = 1/2\) into the formula to get the total sum. If the sum equals 1, then it confirms that the given probability distribution is valid.

Key Concepts

Geometric Series

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For instance, in the sequence 1, 1/2, 1/4, 1/8, ..., the common ratio is 1/2. Each term is half of the term before it.

Mathematically, a geometric series can be expressed as the sum of its terms: \[ S = a + ar + ar^2 + ar^3 + \cdots \]where \( a \) is the first term and \( r \) is the common ratio. The sum of an infinite geometric series converges to a finite value if the common ratio's absolute value is less than one: \[ S = \frac{a}{1 - r} \].
In probability, a geometric series often represents the total probability of mutually exclusive outcomes where each outcome's probability is a constant fraction of the previous outcome's probability.

Random Variable

A random variable, often denoted by \(X\), \(Y\), or \(n\) as in our example, is a variable whose possible values are numerical outcomes of a random phenomenon. There are two types of random variables: discrete and continuous. Discrete random variables have countable outcomes, such as the number of units sold per day.

A probability distribution assigns probabilities to each possible value of the random variable. In the example from the exercise, the probability distribution \(P(n)\) represents the likelihood that \(n\) units are sold on a given day, where \(n\) is a discrete random variable.

Properties of Probabilities

The properties of probabilities define the basics rules that all probability distributions must follow. Firstly, the probability of any specific outcome must fall between 0 and 1, inclusively. Moreover, the probability of all possible outcomes should add up to 1. This is the normalization condition of probabilities.

Another important property is that the probabilities of mutually exclusive events should be summed to find the probability of either event occurring. If the events are not mutually exclusive, the sum needs to be adjusted to account for the overlap.

Summation of Probabilities

The summation of probabilities when dealing with a discrete random variable ensures that the probabilities of all individual outcomes add up to 1. This is a fundamental principle in probability theory, reflecting the certainty that one of the possible outcomes will occur.

In our example, each \(P(n)\) represents the probability of selling \(n\) units. To confirm that the probability distribution is valid, we evaluate the summation of the entire series of probabilities using the geometric series formula. Through this calculation, we demonstrate that indeed, the total sum of probabilities for all possible sales amounts to 1, satisfying the properties of probability distributions.