Question 1193310: Some probability distributions. Here is a probability distribution for a random variable X:
Value of X Probability
4 0.3
5 0.4
6 0.3
(A)Find the mean and standard deviation for this distribution.
(B)Construct a different probability distribution with the same possible values, the same mean, and a larger standard deviation. Show your workand report the standard deviation of your new distribution.
(C)Construct a different probability distribution with the same possible values, the same mean, and a smaller standard deviation. Show your work and report the standard deviation of your new distribution.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Certainly, let's analyze the given probability distribution and construct others with the same mean but different standard deviations.
**A. Mean and Standard Deviation of the Given Distribution**
* **Mean (μ):**
μ = (4 * 0.3) + (5 * 0.4) + (6 * 0.3)
μ = 1.2 + 2 + 1.8
μ = 5
* **Variance (σ²):**
σ² = Σ[(x - μ)² * P(x)]
σ² = [(4 - 5)² * 0.3] + [(5 - 5)² * 0.4] + [(6 - 5)² * 0.3]
σ² = [1 * 0.3] + [0 * 0.4] + [1 * 0.3]
σ² = 0.6
* **Standard Deviation (σ):**
σ = √σ² = √0.6 ≈ 0.7746
**B. Constructing a Distribution with Larger Standard Deviation**
To increase the standard deviation, we need to increase the spread of the probabilities around the mean. We can achieve this by shifting some probability mass away from the mean.
Here's one possible distribution:
* Value of X Probability
4 0.1
5 0.2
6 0.7
* **Calculate the Mean (to verify it's still 5):**
μ = (4 * 0.1) + (5 * 0.2) + (6 * 0.7)
μ = 0.4 + 1 + 4.2
μ = 5
* **Calculate the Variance:**
σ² = [(4 - 5)² * 0.1] + [(5 - 5)² * 0.2] + [(6 - 5)² * 0.7]
σ² = [1 * 0.1] + [0 * 0.2] + [1 * 0.7]
σ² = 0.8
* **Calculate the Standard Deviation:**
σ = √σ² = √0.8 ≈ 0.8944
This distribution has the same mean as the original but a larger standard deviation.
**C. Constructing a Distribution with Smaller Standard Deviation**
To decrease the standard deviation, we need to concentrate the probability mass closer to the mean.
Here's one possible distribution:
* Value of X Probability
4 0.2
5 0.6
6 0.2
* **Calculate the Mean (to verify it's still 5):**
μ = (4 * 0.2) + (5 * 0.6) + (6 * 0.2)
μ = 0.8 + 3 + 1.2
μ = 5
* **Calculate the Variance:**
σ² = [(4 - 5)² * 0.2] + [(5 - 5)² * 0.6] + [(6 - 5)² * 0.2]
σ² = [1 * 0.2] + [0 * 0.6] + [1 * 0.2]
σ² = 0.4
* **Calculate the Standard Deviation:**
σ = √σ² = √0.4 ≈ 0.6325
This distribution has the same mean as the original but a smaller standard deviation.
**Note:** These are just two possible examples. There are many other probability distributions that would satisfy the given conditions.
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