Question

what type of data distribution is shown on the graph?

Explanation:

1. First, recall the characteristics of different data distributions:

• In a symmetric distribution, the data is evenly distributed around the mean, and the left and right sides of the distribution (when graphed as a histogram or a probability mass function graph like here) are mirror - images.
• In a left - skewed (negatively skewed) distribution, the tail of the distribution is longer on the left side (lower values), and the mean is less than the median.
• In a right - skewed (positively skewed) distribution, the tail of the distribution is longer on the right side (higher values), and the mean is greater than the median.

1. Then, analyze the given graph:

• Looking at the graph of \(p_X(x)\) (probability mass function for a discrete random variable \(X\)), we can see that the values of \(X = 0\) and \(X = 4\) have very low probabilities (the bars are very short), and as \(X\) increases from \(1\) to \(3\), the probabilities (heights of the bars) increase. The "tail" of the distribution (the part with lower probabilities) is on the right - hand side (for higher values of \(X\), like \(X = 4\))? Wait, no, wait. Wait, the x - axis values are 0,1,2,3,4. The bar for \(X = 0\) is short, \(X = 1\) is a bit taller, \(X = 2\) taller than \(X = 1\), \(X = 3\) taller than \(X = 2\), and \(X = 4\) is short. Wait, actually, the tail is on the left? No, no. Wait, in a discrete probability distribution, the skewness is determined by the direction of the longer tail. Wait, maybe I made a mistake. Wait, let's re - examine. The bars for \(X = 0\) and \(X = 4\) are short. The bars for \(X = 1\), \(X = 2\), \(X = 3\) are increasing in height. So the left - most values ( \(X = 0\)) have a short bar, and the right - most values ( \(X = 4\)) also have a short bar? Wait, no, maybe it's a right - skewed distribution. Wait, in a right - skewed distribution, the mean is pulled towards the right (higher values) because of the presence of relatively few large values. But in this case, the probabilities for \(X = 1\), \(X = 2\), \(X = 3\) are increasing. Wait, maybe the graph is of a discrete probability distribution, and the shape shows that as \(x\) increases from 0 to 3, the probability \(p_X(x)\) increases, and then at \(x = 4\) it drops. So the tail is on the right (at \(x = 4\)), and the main body of the distribution is on the left - middle. So this is a right - skewed (positively skewed) distribution. Because the tail (the part with lower probability) is on the right side of the graph.

Answer:

Right - Skewed (or Positively Skewed) Distribution