Problem 13 - Calculus for Biology and Medicine

Question

Let \(S=\\{1,2,3, \ldots, 10\\}\), and assume that $$p(k)=\frac{k}{N}, k \in S$$ where \(N\) is a constant. (a) Determine \(N\) so that \(p(k), k \in S\), is a probability mass function. (b) Let \(X\) be a discrete random variable with \(P(X=k)=p(k)\). Find the probability that \(X\) is less than 8 .

Step-by-Step Solution

Answer

(a) \( N = 55 \); (b) \( P(X < 8) = \frac{28}{55} \).

Step 1: Understand the Probability Mass Function

The given function is \( p(k) = \frac{k}{N} \) for each \( k \) in the set \( S = \{1, 2, 3, \ldots, 10\} \). This represents a discrete probability mass function if the probabilities for all \( k \) sum to 1.

Step 2: Set Up the Sum of Probabilities to Equal 1

For \( p(k) \) to be a valid probability mass function, the sum \( \sum_{k=1}^{10} p(k) = 1 \) must hold, where \( p(k) = \frac{k}{N} \). Therefore, we have \[ \frac{1}{N} + \frac{2}{N} + \frac{3}{N} + \ldots + \frac{10}{N} = 1. \]

Step 3: Sum the Series

The series \( 1 + 2 + 3 + \ldots + 10 \) sums to \( \frac{10 \cdot 11}{2} = 55 \). Thus, we rewrite the equation as \[ \frac{55}{N} = 1. \]

Step 4: Solve for N

To find \( N \), solve \( \frac{55}{N} = 1 \) to get \( N = 55 \). This ensures \( p(k) = \frac{k}{55} \) is a valid probability mass function, as the sum is 1.

Step 5: Probability That X is Less Than 8

The probability that \( X \) is less than 8 includes events from \( X = 1 \) to \( X = 7 \). We need to find \( P(X < 8) = \sum_{k=1}^{7} p(k) = \sum_{k=1}^{7} \frac{k}{55}. \)

Step 6: Calculate the Cumulative Probability

Calculate \( \sum_{k=1}^{7} k = 1 + 2 + 3 + 4 + 5 + 6 + 7 = \frac{7 \cdot 8}{2} = 28 \). Thus, \[ P(X < 8) = \frac{28}{55}. \]

Key Concepts

Discrete Random Variable

In probability and statistics, a **discrete random variable** is a type of random variable that can take on a countable number of distinct values. This means that the set of possible outcomes can be listed, often as integers. Examples include rolling a die or the number of emails received in a day.
For this exercise, the random variable \(X\) can take any value in the set \(S = \{1, 2, 3, \ldots, 10\}\). Each possible value of \(X\) has a probability associated with it, given by the function \(p(k) = \frac{k}{N}\).

• The outcomes are finite and therefore countable.
• Probabilities are assigned based on a formula, reflecting how the random variable behaves.

Understanding what a discrete random variable is will help solve and make sense of the probability calculations presented in this problem.

Sum of Probabilities

The **sum of probabilities** for all possible outcomes of a discrete random variable must equal 1. This is a fundamental property of any probability distribution.
For the given exercise, we need \(\sum_{k=1}^{10} p(k) = 1\). The formula \(p(k) = \frac{k}{N}\) needs the correct constant \(N\) such that this condition holds.

• Each individual probability is calculated by substituting \(k\) into the function \(p(k)\).
• Once calculated, ensure their total equals 1 to validate it as a probability mass function (PMF).

In our problem, solving for \(N\), we find that \(N = 55\) is necessary so that when all probabilities are summed, they satisfy the condition \(\frac{55}{N} = 1\).

Cumulative Probability

**Cumulative probability** refers to the probability that the value of a discrete random variable is less than or equal to a certain value. It helps determine the likelihood of multiple events happening up to a point.
In this exercise, we calculate the probability that \(X\) is less than 8. This involves summing up the probabilities for \(X\) being values 1 through 7.

• Calculate each individual probability from \(X = 1\) to \(X = 7\).
• Add these probabilities together to find the cumulative probability.

In this case, we sum these probabilities: \(\sum_{k=1}^{7} p(k)\), resulting in the cumulative probability \(P(X < 8) = \frac{28}{55}\). This gives insight into the likelihood of outcomes falling within this range.