\( \left. \begin{array} { c c c c c c c c } { X } & { 1 } & { 2 } & { 3 } & { 4 } & { 5 } & { 6 } & { 7 } \\ { P ( X ) } & { \frac { 1 } { 7 } } & { \frac { 1 } { 7 } } & { \frac { 1 } { 7 } } & { \frac { 1 } { 7 } } & { \frac { 1 } { 7 } } & { \frac { 1 } { 7 } } & { \frac { 1 } { 7 } } \end{array} \right. \)

The table you've provided represents a **discrete uniform probability distribution** for the random variable \( X \). Let's break down and analyze this distribution in detail. ### Given Probability Distribution \[ \begin{array}{c|ccccccc} X & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline P(X) & \frac{1}{7} & \frac{1}{7} & \frac{1}{7} & \frac{1}{7} & \frac{1}{7} & \frac{1}{7} & \frac{1}{7} \\ \end{array} \] ### Key Characteristics 1. **Discrete Uniform Distribution**: - **Discrete**: \( X \) takes on a finite number of distinct values. - **Uniform**: Each outcome has an equal probability of occurrence. 2. **Support of \( X \)**: - The random variable \( X \) can take any integer value from 1 to 7, inclusive. 3. **Probability Mass Function (PMF)**: - For each \( x \in \{1, 2, 3, 4, 5, 6, 7\} \), \( P(X = x) = \frac{1}{7} \). ### Verification of Probability Distribution To confirm that this is a valid probability distribution, the sum of all probabilities should equal 1: \[ \sum_{x=1}^{7} P(X = x) = 7 \times \frac{1}{7} = 1 \] This satisfies the requirement for a probability distribution. ### Calculating Key Statistics 1. **Expected Value (Mean)**: The expected value \( E(X) \) of a discrete uniform distribution is given by: \[ E(X) = \frac{a + b}{2} \] where \( a \) and \( b \) are the minimum and maximum values of \( X \), respectively. For this distribution: \[ E(X) = \frac{1 + 7}{2} = 4 \] 2. **Variance**: The variance \( \text{Var}(X) \) measures the spread of the distribution and is given by: \[ \text{Var}(X) = \frac{(b - a + 1)^2 - 1}{12} \] Plugging in the values: \[ \text{Var}(X) = \frac{(7 - 1 + 1)^2 - 1}{12} = \frac{7^2 - 1}{12} = \frac{49 - 1}{12} = \frac{48}{12} = 4 \] 3. **Standard Deviation**: The standard deviation \( \sigma_X \) is the square root of the variance: \[ \sigma_X = \sqrt{\text{Var}(X)} = \sqrt{4} = 2 \] 4. **Probability Generating Function (PGF)**: The PGF \( G_X(t) \) for a discrete random variable is: \[ G_X(t) = E(t^X) = \sum_{x=1}^{7} P(X = x) t^x = \frac{1}{7}(t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7) \] ### Applications and Examples Such uniform distributions are common in scenarios where each outcome is equally likely. Examples include: - **Rolling a Fair Die**: A standard die has outcomes \( \{1, 2, 3, 4, 5, 6\} \), each with probability \( \frac{1}{6} \). Your distribution extends this idea to seven outcomes. - **Random Selection**: Picking a random day from a week (if considering an extra day or specific scenarios). ### Potential Questions and Extensions Depending on your interest or application, here are some extensions or related questions you might consider: 1. **Cumulative Distribution Function (CDF)**: - Calculate \( P(X \leq x) \) for each \( x \). 2. **Moment Generating Function (MGF)**: - Determine the MGF and use it to find moments of \( X \). 3. **Comparisons with Other Distributions**: - Compare the uniform distribution with other discrete distributions like binomial or Poisson. 4. **Applications in Real-World Scenarios**: - Explore how this distribution can model real-life situations where outcomes are equally likely. If you have a specific question or need further assistance regarding this probability distribution, feel free to ask!