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Chapter 3: Problem 11

For Exercises 3.1 .10 to \(3.1 .12,\) verify that the following functions are probability mass functions, and determine the requested probabilities. $$ \begin{aligned} &\text f(x)=\frac{2 x+1}{25}, \quad x=0,1,2,3,4\\\ &\begin{array}{ll} \text { a. } P(X=4) & \text { b. } P(X \leq 1) \\ text { c. } P(2 \leq X<4) & \text { d. } P(X>-10) \end{array} \end{aligned} $$

Short Answer

Expert verified
P(X=4) = 9/25, P(X 鈮� 1) = 4/25, P(2 鈮� X < 4) = 12/25, P(X > -10) = 1.

Step by step solution

01

Verify the Probability Mass Function (PMF)

To ensure the function \( f(x) = \frac{2x+1}{25} \) is a valid PMF, we need to check two conditions: 1) \( f(x) \geq 0 \) for all \( x \), and 2) the sum of \( f(x) \) across all possible values is equal to 1. For \( x = 0, 1, 2, 3, 4 \), calculate \( f(x) \): - \( f(0) = \frac{2(0)+1}{25} = \frac{1}{25} \)- \( f(1) = \frac{2(1)+1}{25} = \frac{3}{25} \)- \( f(2) = \frac{2(2)+1}{25} = \frac{5}{25} \)- \( f(3) = \frac{2(3)+1}{25} = \frac{7}{25} \)- \( f(4) = \frac{2(4)+1}{25} = \frac{9}{25} \)Now, sum them up:\( \frac{1}{25} + \frac{3}{25} + \frac{5}{25} + \frac{7}{25} + \frac{9}{25} = \frac{25}{25} = 1 \).Both conditions are satisfied, verifying that \( f(x) \) is a PMF.
02

Find P(X=4)

Using the PMF, calculate the probability when \(X = 4\):\[ P(X=4) = f(4) = \frac{2(4)+1}{25} = \frac{9}{25} \]
03

Calculate P(X 鈮� 1)

Calculate the cumulative probability for \(X \leq 1\):\[ P(X \leq 1) = P(X=0) + P(X=1) = \frac{1}{25} + \frac{3}{25} = \frac{4}{25} \]
04

Determine P(2 鈮� X < 4)

Find the probability of \(X\) being between 2 and 4. Since \(X\) is discrete, we're actually looking at \(2 \leq X < 4\), which includes \(X = 2\) and \(X = 3\):\[ P(2 \leq X < 4) = P(X=2) + P(X=3) = \frac{5}{25} + \frac{7}{25} = \frac{12}{25} \]
05

Evaluate P(X > -10)

Since all possible values of \(X\) (0, 1, 2, 3, 4) are greater than -10, the probability is simply the sum of all probabilities in the PMF, which is 1:\[ P(X > -10) = 1 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Discrete Probability Distribution
A discrete probability distribution is a statistical function that defines the probabilities of outcomes for a random variable that can take on discrete values. These are countable and often integers. In our current context, the random variable \(X\) can take the values \(0, 1, 2, 3,\) and \(4\).

To represent these probabilities, we use a probability mass function (PMF). The PMF provides a compact and convenient way to express the likelihood of each outcome.

Characteristics of a discrete probability distribution:
  • The sum of all probabilities in the distribution must equal 1. This ensures that one of the defined outcomes will certainly occur.
  • Each value of the random variable has a unique probability associated with it, which must be non-negative.
Cumulative Probability
Cumulative probability refers to the probability that a random variable will take a value less than or equal to a specific value.

In simpler terms, it's about accumulating probabilities. For instance, when we calculated \( P(X \leq 1) \), we determined the chances that \(X\) is 0 or 1. The cumulative probability is computed by summing the individual probabilities from the PMF:

\[ P(X \leq 1) = P(X=0) + P(X=1) = \frac{1}{25} + \frac{3}{25} = \frac{4}{25} \]

Benefits of cumulative probability include giving an overall picture of likelihoods up to a certain point, which is especially useful in determining percentiles and ranges in various analyses.
Calculating Probability
Calculating probability using a PMF involves directly plugging in the values of the random variable into the function.

For example, to find the probability that \(X=4\), substitute 4 into the PMF:

\[ P(X=4) = f(4) = \frac{2 \times 4 + 1}{25} = \frac{9}{25} \]

Another type includes calculating probabilities over ranges. For instance, \(P(2 \leq X < 4)\) requires adding probabilities at \(X = 2\) and \(X = 3\):
  • \(P(X=2) = \frac{5}{25}\)
  • \(P(X=3) = \frac{7}{25}\)
  • So, \(P(2 \leq X < 4) = \frac{5}{25} + \frac{7}{25} = \frac{12}{25}\)
Verification of PMF
The verification of a probability mass function (PMF) ensures it is both valid and reliable for use in probability calculations.

For any PMF, two conditions must be verified:
  • Each probability \(f(x)\) must be non-negative for each \(x\) in its possible range.
  • The sum of all probabilities across the range must equal 1: \( \sum f(x) = 1 \)
In our example, the PMF was checked and found valid as:
- All calculated values, \( \frac{1}{25}, \frac{3}{25}, \frac{5}{25}, \frac{7}{25}, \frac{9}{25} \), are non-negative.
- Summing these gives \( \frac{25}{25} = 1 \), meeting the second condition. This confirms the PMF is correctly defined.

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Most popular questions from this chapter

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