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Compute the mean and standard deviation of the random variable with the given discrete probability distribution. x| -4 -1 6 8 9...Question
Compute the mean and standard deviation of the random variable with the given discrete probability distribution.
x| -5, 1, 4, 6, 8
P(x)| 0.11, 0.26, 0.23, 0.28, 0.12
(a) Find the mean. Round the answer to three decimal places, if necessary.
(b) Find the standard deviation. Round the answer to three decimal places, if necessary.
x| -5, 1, 4, 6, 8
P(x)| 0.11, 0.26, 0.23, 0.28, 0.12
(a) Find the mean. Round the answer to three decimal places, if necessary.
(b) Find the standard deviation. Round the answer to three decimal places, if necessary.
Answers
Answer
Both answers are incorrect..
Answer
what is the square root of 15.68
Answer
As answer is the mean is 3.27.
Now find the standard deviation for B
Now find the standard deviation for B
Answered by
GPT-4o mini
To find the mean and standard deviation of the discrete random variable given its probability distribution, we can use the following formulas:
### Mean (µ):
The mean of a discrete probability distribution is calculated using the formula:
\[
\mu = \sum (x \cdot P(x))
\]
### Standard Deviation (σ):
The standard deviation is calculated using the formula:
\[
\sigma = \sqrt{\sum (x^2 \cdot P(x)) - \mu^2}
\]
### Given Data:
Let's summarize the provided data:
\[
\begin{array}{|c|c|}
\hline
x & P(x) \\
\hline
-5 & 0.11 \\
1 & 0.26 \\
4 & 0.23 \\
6 & 0.28 \\
8 & 0.12 \\
\hline
\end{array}
\]
### (a) Calculate the Mean (µ):
First, we compute \(x \cdot P(x)\) for each value:
1. \(x = -5\): \(-5 \cdot 0.11 = -0.55\)
2. \(x = 1\): \(1 \cdot 0.26 = 0.26\)
3. \(x = 4\): \(4 \cdot 0.23 = 0.92\)
4. \(x = 6\): \(6 \cdot 0.28 = 1.68\)
5. \(x = 8\): \(8 \cdot 0.12 = 0.96\)
Now sum these products:
\[
\mu = -0.55 + 0.26 + 0.92 + 1.68 + 0.96 = 2.27
\]
### (b) Calculate the Standard Deviation (σ):
Next, we compute \(x^2 \cdot P(x)\):
1. \(x = -5\): \((-5)^2 \cdot 0.11 = 25 \cdot 0.11 = 2.75\)
2. \(x = 1\): \(1^2 \cdot 0.26 = 1 \cdot 0.26 = 0.26\)
3. \(x = 4\): \(4^2 \cdot 0.23 = 16 \cdot 0.23 = 3.68\)
4. \(x = 6\): \(6^2 \cdot 0.28 = 36 \cdot 0.28 = 10.08\)
5. \(x = 8\): \(8^2 \cdot 0.12 = 64 \cdot 0.12 = 7.68\)
Now sum these products:
\[
\sum (x^2 \cdot P(x)) = 2.75 + 0.26 + 3.68 + 10.08 + 7.68 = 24.45
\]
Now we can compute the variance:
\[
\sigma^2 = \sum (x^2 \cdot P(x)) - \mu^2 = 24.45 - (2.27)^2
\]
\[
(2.27)^2 = 5.1529
\]
\[
\sigma^2 = 24.45 - 5.1529 = 19.2971
\]
Now take the square root to find the standard deviation:
\[
\sigma = \sqrt{19.2971} \approx 4.39
\]
### Final Answers:
(a) Mean (µ): **2.270**
(b) Standard Deviation (σ): **4.390**
### Mean (µ):
The mean of a discrete probability distribution is calculated using the formula:
\[
\mu = \sum (x \cdot P(x))
\]
### Standard Deviation (σ):
The standard deviation is calculated using the formula:
\[
\sigma = \sqrt{\sum (x^2 \cdot P(x)) - \mu^2}
\]
### Given Data:
Let's summarize the provided data:
\[
\begin{array}{|c|c|}
\hline
x & P(x) \\
\hline
-5 & 0.11 \\
1 & 0.26 \\
4 & 0.23 \\
6 & 0.28 \\
8 & 0.12 \\
\hline
\end{array}
\]
### (a) Calculate the Mean (µ):
First, we compute \(x \cdot P(x)\) for each value:
1. \(x = -5\): \(-5 \cdot 0.11 = -0.55\)
2. \(x = 1\): \(1 \cdot 0.26 = 0.26\)
3. \(x = 4\): \(4 \cdot 0.23 = 0.92\)
4. \(x = 6\): \(6 \cdot 0.28 = 1.68\)
5. \(x = 8\): \(8 \cdot 0.12 = 0.96\)
Now sum these products:
\[
\mu = -0.55 + 0.26 + 0.92 + 1.68 + 0.96 = 2.27
\]
### (b) Calculate the Standard Deviation (σ):
Next, we compute \(x^2 \cdot P(x)\):
1. \(x = -5\): \((-5)^2 \cdot 0.11 = 25 \cdot 0.11 = 2.75\)
2. \(x = 1\): \(1^2 \cdot 0.26 = 1 \cdot 0.26 = 0.26\)
3. \(x = 4\): \(4^2 \cdot 0.23 = 16 \cdot 0.23 = 3.68\)
4. \(x = 6\): \(6^2 \cdot 0.28 = 36 \cdot 0.28 = 10.08\)
5. \(x = 8\): \(8^2 \cdot 0.12 = 64 \cdot 0.12 = 7.68\)
Now sum these products:
\[
\sum (x^2 \cdot P(x)) = 2.75 + 0.26 + 3.68 + 10.08 + 7.68 = 24.45
\]
Now we can compute the variance:
\[
\sigma^2 = \sum (x^2 \cdot P(x)) - \mu^2 = 24.45 - (2.27)^2
\]
\[
(2.27)^2 = 5.1529
\]
\[
\sigma^2 = 24.45 - 5.1529 = 19.2971
\]
Now take the square root to find the standard deviation:
\[
\sigma = \sqrt{19.2971} \approx 4.39
\]
### Final Answers:
(a) Mean (µ): **2.270**
(b) Standard Deviation (σ): **4.390**
Answered by
GPT-4o mini
Let's go through the calculations step-by-step to ensure we get the correct results for the mean and standard deviation of the given discrete probability distribution.
### Given Data:
\[
\begin{array}{|c|c|}
\hline
x & P(x) \\
\hline
-5 & 0.11 \\
1 & 0.26 \\
4 & 0.23 \\
6 & 0.28 \\
8 & 0.12 \\
\hline
\end{array}
\]
### (a) Calculate the Mean (µ):
Using the formula for the mean:
\[
\mu = \sum (x \cdot P(x))
\]
We will compute each \( x \cdot P(x) \):
1. For \( x = -5 \):
\[
-5 \cdot 0.11 = -0.55
\]
2. For \( x = 1 \):
\[
1 \cdot 0.26 = 0.26
\]
3. For \( x = 4 \):
\[
4 \cdot 0.23 = 0.92
\]
4. For \( x = 6 \):
\[
6 \cdot 0.28 = 1.68
\]
5. For \( x = 8 \):
\[
8 \cdot 0.12 = 0.96
\]
Now, we sum these results:
\[
\mu = -0.55 + 0.26 + 0.92 + 1.68 + 0.96 = 2.27
\]
Thus, the mean \(\mu\) is:
\[
\mu = 2.27
\]
### (b) Calculate the Standard Deviation (σ):
To calculate the standard deviation, we first need to compute \( x^2 \cdot P(x) \):
1. For \( x = -5 \):
\[
(-5)^2 \cdot 0.11 = 25 \cdot 0.11 = 2.75
\]
2. For \( x = 1 \):
\[
1^2 \cdot 0.26 = 1 \cdot 0.26 = 0.26
\]
3. For \( x = 4 \):
\[
4^2 \cdot 0.23 = 16 \cdot 0.23 = 3.68
\]
4. For \( x = 6 \):
\[
6^2 \cdot 0.28 = 36 \cdot 0.28 = 10.08
\]
5. For \( x = 8 \):
\[
8^2 \cdot 0.12 = 64 \cdot 0.12 = 7.68
\]
Now sum these products:
\[
\sum (x^2 \cdot P(x)) = 2.75 + 0.26 + 3.68 + 10.08 + 7.68 = 24.45
\]
Calculate the variance using:
\[
\sigma^2 = \sum (x^2 \cdot P(x)) - \mu^2
\]
Calculating \( \mu^2 \):
\[
\mu^2 = (2.27)^2 = 5.1529
\]
Now calculate variance:
\[
\sigma^2 = 24.45 - 5.1529 = 19.2971
\]
Finally, taking the square root gives us the standard deviation:
\[
\sigma = \sqrt{19.2971} \approx 4.39
\]
### Correct Final Answers
1. (a) Mean (µ): **2.270**
2. (b) Standard Deviation (σ): **4.390**
If you could provide the expected values, we can also cross-check if any assumptions of the input data were misinterpreted.
### Given Data:
\[
\begin{array}{|c|c|}
\hline
x & P(x) \\
\hline
-5 & 0.11 \\
1 & 0.26 \\
4 & 0.23 \\
6 & 0.28 \\
8 & 0.12 \\
\hline
\end{array}
\]
### (a) Calculate the Mean (µ):
Using the formula for the mean:
\[
\mu = \sum (x \cdot P(x))
\]
We will compute each \( x \cdot P(x) \):
1. For \( x = -5 \):
\[
-5 \cdot 0.11 = -0.55
\]
2. For \( x = 1 \):
\[
1 \cdot 0.26 = 0.26
\]
3. For \( x = 4 \):
\[
4 \cdot 0.23 = 0.92
\]
4. For \( x = 6 \):
\[
6 \cdot 0.28 = 1.68
\]
5. For \( x = 8 \):
\[
8 \cdot 0.12 = 0.96
\]
Now, we sum these results:
\[
\mu = -0.55 + 0.26 + 0.92 + 1.68 + 0.96 = 2.27
\]
Thus, the mean \(\mu\) is:
\[
\mu = 2.27
\]
### (b) Calculate the Standard Deviation (σ):
To calculate the standard deviation, we first need to compute \( x^2 \cdot P(x) \):
1. For \( x = -5 \):
\[
(-5)^2 \cdot 0.11 = 25 \cdot 0.11 = 2.75
\]
2. For \( x = 1 \):
\[
1^2 \cdot 0.26 = 1 \cdot 0.26 = 0.26
\]
3. For \( x = 4 \):
\[
4^2 \cdot 0.23 = 16 \cdot 0.23 = 3.68
\]
4. For \( x = 6 \):
\[
6^2 \cdot 0.28 = 36 \cdot 0.28 = 10.08
\]
5. For \( x = 8 \):
\[
8^2 \cdot 0.12 = 64 \cdot 0.12 = 7.68
\]
Now sum these products:
\[
\sum (x^2 \cdot P(x)) = 2.75 + 0.26 + 3.68 + 10.08 + 7.68 = 24.45
\]
Calculate the variance using:
\[
\sigma^2 = \sum (x^2 \cdot P(x)) - \mu^2
\]
Calculating \( \mu^2 \):
\[
\mu^2 = (2.27)^2 = 5.1529
\]
Now calculate variance:
\[
\sigma^2 = 24.45 - 5.1529 = 19.2971
\]
Finally, taking the square root gives us the standard deviation:
\[
\sigma = \sqrt{19.2971} \approx 4.39
\]
### Correct Final Answers
1. (a) Mean (µ): **2.270**
2. (b) Standard Deviation (σ): **4.390**
If you could provide the expected values, we can also cross-check if any assumptions of the input data were misinterpreted.
Answered by
GPT-4o mini
The square root of 15.68 is approximately:
\[
\sqrt{15.68} \approx 3.96
\]
If you need a more precise value, it's:
\[
\sqrt{15.68} \approx 3.96
\]
So the value rounded to two decimal places is **3.96**.
\[
\sqrt{15.68} \approx 3.96
\]
If you need a more precise value, it's:
\[
\sqrt{15.68} \approx 3.96
\]
So the value rounded to two decimal places is **3.96**.
Answered by
GPT-4o mini
Let's first summarize the calculations again based on the corrected mean of \( \mu = 3.27 \). We will compute the standard deviation using the provided probabilities.
### Given Data:
\[
\begin{array}{|c|c|}
\hline
x & P(x) \\
\hline
-5 & 0.11 \\
1 & 0.26 \\
4 & 0.23 \\
6 & 0.28 \\
8 & 0.12 \\
\hline
\end{array}
\]
### Calculate Standard Deviation (σ):
We use the formula for standard deviation:
\[
\sigma = \sqrt{\sum (x^2 \cdot P(x)) - \mu^2}
\]
#### Step 1: Calculate \( x^2 \cdot P(x) \)
1. For \( x = -5 \):
\[
(-5)^2 \cdot 0.11 = 25 \cdot 0.11 = 2.75
\]
2. For \( x = 1 \):
\[
1^2 \cdot 0.26 = 1 \cdot 0.26 = 0.26
\]
3. For \( x = 4 \):
\[
4^2 \cdot 0.23 = 16 \cdot 0.23 = 3.68
\]
4. For \( x = 6 \):
\[
6^2 \cdot 0.28 = 36 \cdot 0.28 = 10.08
\]
5. For \( x = 8 \):
\[
8^2 \cdot 0.12 = 64 \cdot 0.12 = 7.68
\]
#### Step 2: Sum these products
\[
\sum (x^2 \cdot P(x)) = 2.75 + 0.26 + 3.68 + 10.08 + 7.68 = 24.45
\]
#### Step 3: Calculate \( \mu^2 \)
Given \( \mu = 3.27 \):
\[
\mu^2 = (3.27)^2 = 10.7129
\]
#### Step 4: Calculate Variance
\[
\sigma^2 = \sum (x^2 \cdot P(x)) - \mu^2 = 24.45 - 10.7129 = 13.7371
\]
#### Step 5: Calculate Standard Deviation
Now take the square root to find the standard deviation:
\[
\sigma = \sqrt{13.7371} \approx 3.71
\]
### Final Answer
The standard deviation (σ) is approximately **3.710** when rounded to three decimal places.
### Given Data:
\[
\begin{array}{|c|c|}
\hline
x & P(x) \\
\hline
-5 & 0.11 \\
1 & 0.26 \\
4 & 0.23 \\
6 & 0.28 \\
8 & 0.12 \\
\hline
\end{array}
\]
### Calculate Standard Deviation (σ):
We use the formula for standard deviation:
\[
\sigma = \sqrt{\sum (x^2 \cdot P(x)) - \mu^2}
\]
#### Step 1: Calculate \( x^2 \cdot P(x) \)
1. For \( x = -5 \):
\[
(-5)^2 \cdot 0.11 = 25 \cdot 0.11 = 2.75
\]
2. For \( x = 1 \):
\[
1^2 \cdot 0.26 = 1 \cdot 0.26 = 0.26
\]
3. For \( x = 4 \):
\[
4^2 \cdot 0.23 = 16 \cdot 0.23 = 3.68
\]
4. For \( x = 6 \):
\[
6^2 \cdot 0.28 = 36 \cdot 0.28 = 10.08
\]
5. For \( x = 8 \):
\[
8^2 \cdot 0.12 = 64 \cdot 0.12 = 7.68
\]
#### Step 2: Sum these products
\[
\sum (x^2 \cdot P(x)) = 2.75 + 0.26 + 3.68 + 10.08 + 7.68 = 24.45
\]
#### Step 3: Calculate \( \mu^2 \)
Given \( \mu = 3.27 \):
\[
\mu^2 = (3.27)^2 = 10.7129
\]
#### Step 4: Calculate Variance
\[
\sigma^2 = \sum (x^2 \cdot P(x)) - \mu^2 = 24.45 - 10.7129 = 13.7371
\]
#### Step 5: Calculate Standard Deviation
Now take the square root to find the standard deviation:
\[
\sigma = \sqrt{13.7371} \approx 3.71
\]
### Final Answer
The standard deviation (σ) is approximately **3.710** when rounded to three decimal places.
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