Math Problem Statement

In the probability distribution to the​ right, the random variable X represents the number of hits a baseball player obtained in a game over the course of a season. Complete parts​ (a) through​ (f) below. x ​P(x)

0 0.1663 1 0.3353 2 0.2865 3 0.1484 4 0.0381 5 0.0254 Question content area bottom Part 1 ​(a) Verify that this is a discrete probability distribution. This is a discrete probability distribution because all of the probabilities are between    0 and    1​, ​inclusive, and the sum of the probabilities is    1. ​(Type whole numbers. Use ascending​ order.) Part 2 ​(b) Draw a graph of the probability distribution. Describe the shape of the distribution. Graph the probability distribution. Choose the correct graph below. A. 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 Number of Hits Probability

The graph of a probability distribution has a horizontal x-axis labeled "Number of Hits" from 0 to 5 in intervals of 1 and a vertical y-axis labeled "Probability" from 0 to 0.4 in intervals of 0.05. Vertical line segments are centered on each of the horizontal axis tick marks. The approximate heights of the vertical line segments are as follows, with the horizontal coordinate listed first and the line height listed second: 0, 0.03; 1, 0.04; 2, 0.15; 3, 0.29; 4, 0.34; 5, 0.17. B. 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 Number of Hits Probability

The graph of a probability distribution has a horizontal x-axis labeled "Number of Hits" from 0 to 5 in intervals of 1 and a vertical y-axis labeled "Probability" from 0 to 0.4 in intervals of 0.05. Vertical line segments are centered on each of the horizontal axis tick marks. The approximate heights of the vertical line segments are as follows, with the horizontal coordinate listed first and the line height listed second: 0, 0.15; 1, 0.04; 2, 0.03; 3, 0.17; 4, 0.34; 5, 0.29. C. 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 Number of Hits Probability

The graph of a probability distribution has a horizontal x-axis labeled "Number of Hits" from 0 to 5 in intervals of 1 and a vertical y-axis labeled "Probability" from 0 to 0.4 in intervals of 0.05. Vertical line segments are centered on each of the horizontal axis tick marks. The approximate heights of the vertical line segments are as follows, with the horizontal coordinate listed first and the line height listed second: 0.34; 1, 0.15; 2, 0.03; 3, 0.17; 4, 0.29; 5, 0.04. D. 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 Number of Hits Probability

The graph of a probability distribution has a horizontal x-axis labeled "Number of Hits" from 0 to 5 in intervals of 1 and a vertical y-axis labeled "Probability" from 0 to 0.4 in intervals of 0.05. Vertical line segments are centered on each of the horizontal axis tick marks. The approximate heights of the vertical line segments are as follows, with the horizontal coordinate listed first and the line height listed second: 0, 0.17; 1, 0.34; 2, 0.29; 3, 0.15; 4, 0.04; 5, 0.03. Your answer is correct. Part 3 Describe the shape of the distribution. The distribution has one mode and is skewed right. Part 4 ​(c) Compute and interpret the mean of the random variable X. mu Subscript Upper Xequals    enter your response here hits ​(Type an integer or a decimal. Do not​ round.)

Solution

Part 1 (a)

To verify that this is a discrete probability distribution, we need to check two conditions:

  1. Each probability is between 0 and 1, inclusive:

    The probabilities given are:

    • P(0)=0.1663P(0) = 0.1663
    • P(1)=0.3353P(1) = 0.3353
    • P(2)=0.2865P(2) = 0.2865
    • P(3)=0.1484P(3) = 0.1484
    • P(4)=0.0381P(4) = 0.0381
    • P(5)=0.0254P(5) = 0.0254

    All of these probabilities lie between 0 and 1.

  2. The sum of the probabilities is equal to 1: 0.1663+0.3353+0.2865+0.1484+0.0381+0.0254=10.1663 + 0.3353 + 0.2865 + 0.1484 + 0.0381 + 0.0254 = 1 Therefore, this is a discrete probability distribution because the probabilities are between 0 and 1, and their sum is 1.

Part 2 (b)

To determine the correct graph, let's match the values given for each P(x)P(x):

  • P(0)=0.1663P(0) = 0.1663
  • P(1)=0.3353P(1) = 0.3353
  • P(2)=0.2865P(2) = 0.2865
  • P(3)=0.1484P(3) = 0.1484
  • P(4)=0.0381P(4) = 0.0381
  • P(5)=0.0254P(5) = 0.0254

The correct graph is D because the heights of the bars closely match the probabilities. The heights for P(0),P(1),P(2),P(3),P(4),P(0), P(1), P(2), P(3), P(4), and P(5)P(5) are accurately represented in this option.

Part 3

The shape of the distribution can be described as:

  • Unimodal (has one mode, with the peak at x=1x = 1).
  • Skewed to the right because there is a tail extending towards the higher number of hits (3, 4, 5).

Part 4 (c)

To compute the mean (expected value) of the random variable XX, we use the formula:

μX=∑[x⋅P(x)]\mu_X = \sum [x \cdot P(x)]

μX=(0⋅0.1663)+(1⋅0.3353)+(2⋅0.2865)+(3⋅0.1484)+(4⋅0.0381)+(5⋅0.0254)\mu_X = (0 \cdot 0.1663) + (1 \cdot 0.3353) + (2 \cdot 0.2865) + (3 \cdot 0.1484) + (4 \cdot 0.0381) + (5 \cdot 0.0254)

μX=0+0.3353+0.5730+0.4452+0.1524+0.1270=1.633\mu_X = 0 + 0.3353 + 0.5730 + 0.4452 + 0.1524 + 0.1270 = 1.633

So, the mean number of hits μX=1.633\mu_X = 1.633.

Interpretation: The mean number of hits per game for the player over the course of the season is approximately 1.633 hits per game.

Would you like more details on any part? Here are five related questions for further exploration:

  1. How would you calculate the variance and standard deviation of this distribution?
  2. What is the probability that the player gets at least 2 hits in a game?
  3. How would you describe the median of this distribution?
  4. If you were to simulate this data, what would a typical sequence of hits look like?
  5. How does skewness affect the interpretation of the average number of hits?

Tip: When dealing with probability distributions, always ensure that the total sum of probabilities equals 1 to confirm the validity of the distribution.

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Math Problem Analysis

Mathematical Concepts

Probability Distribution
Expected Value (Mean)
Discrete Probability Distribution
Skewness

Formulas

Sum of probabilities in a discrete distribution: P(x_1) + P(x_2) + ... + P(x_n) = 1
Mean (Expected value) formula: μ_X = Σ [x * P(x)]

Theorems

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Suitable Grade Level

Grades 9-12

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