Find trusted answers to your questions with the help of IDNLearn.com's knowledgeable community. Ask your questions and receive reliable and comprehensive answers from our dedicated community of professionals.
Sagot :
To find the mean and standard deviation of the random variable [tex]\( x \)[/tex], we will follow these steps:
### Mean
The mean (or expected value) of a random variable [tex]\( x \)[/tex] is calculated using the formula:
[tex]\[ \mu = \sum (x_i \cdot P(x_i)) \][/tex]
where [tex]\( x_i \)[/tex] are the values of the random variable and [tex]\( P(x_i) \)[/tex] are the corresponding probabilities.
Given the values:
- [tex]\( x \)[/tex] = [tex]\( \{0, 1, 2, 3, 4\} \)[/tex]
- [tex]\( P(x) \)[/tex] = [tex]\( \left\{\frac{3}{17}, \frac{5}{17}, \frac{6}{17}, \frac{2}{17}, \frac{1}{17} \right\} \)[/tex]
We calculate the mean as follows:
[tex]\[ \mu = (0 \cdot \frac{3}{17}) + (1 \cdot \frac{5}{17}) + (2 \cdot \frac{6}{17}) + (3 \cdot \frac{2}{17}) + (4 \cdot \frac{1}{17}) \][/tex]
[tex]\[ \mu = 0 + \left(\frac{5}{17}\right) + \left(\frac{12}{17}\right) + \left(\frac{6}{17}\right) + \left(\frac{4}{17}\right) \][/tex]
[tex]\[ \mu = \frac{5 + 12 + 6 + 4}{17} = \frac{27}{17} \approx 1.59 \][/tex]
So, the mean is:
[tex]\[ \text{mean} = 1.59 \][/tex]
### Standard Deviation
The standard deviation is the square root of the variance. The variance ([tex]\( \sigma^2 \)[/tex]) is calculated using the formula:
[tex]\[ \sigma^2 = \sum (P(x_i) \cdot (x_i - \mu)^2) \][/tex]
To find the variance, we need the values of [tex]\( (x_i - \mu)^2 \)[/tex]:
[tex]\[ (0 - 1.59)^2 = 2.5281 \][/tex]
[tex]\[ (1 - 1.59)^2 = 0.3481 \][/tex]
[tex]\[ (2 - 1.59)^2 = 0.1681 \][/tex]
[tex]\[ (3 - 1.59)^2 = 2.0161 \][/tex]
[tex]\[ (4 - 1.59)^2 = 5.8561 \][/tex]
Now we calculate the variance:
[tex]\[ \sigma^2 = ( \frac{3}{17} \cdot 2.5281 ) + ( \frac{5}{17} \cdot 0.3481 ) + ( \frac{6}{17} \cdot 0.1681 ) + ( \frac{2}{17} \cdot 2.0161 ) + ( \frac{1}{17} \cdot 5.8561 ) \][/tex]
[tex]\[ \sigma^2 = \left( \frac{7.5843}{17} \right) + \left( \frac{1.7405}{17} \right) + \left( \frac{1.0086}{17} \right) + \left( \frac{4.0322}{17} \right) + \left( \frac{5.8561}{17} \right) \][/tex]
[tex]\[ \sigma^2 = \frac{7.5843 + 1.7405 + 1.0086 + 4.0322 + 5.8561}{17} = \frac{20.2217}{17} \approx 1.189 \][/tex]
The standard deviation is the square root of the variance:
[tex]\[ \sigma = \sqrt{1.189} \approx 1.09 \][/tex]
So, the standard deviation is:
[tex]\[ \text{standard deviation} = 1.09 \][/tex]
### Answers
a) Mean [tex]\( = 1.59 \)[/tex]
b) Standard Deviation [tex]\( = 1.09 \)[/tex]
### Mean
The mean (or expected value) of a random variable [tex]\( x \)[/tex] is calculated using the formula:
[tex]\[ \mu = \sum (x_i \cdot P(x_i)) \][/tex]
where [tex]\( x_i \)[/tex] are the values of the random variable and [tex]\( P(x_i) \)[/tex] are the corresponding probabilities.
Given the values:
- [tex]\( x \)[/tex] = [tex]\( \{0, 1, 2, 3, 4\} \)[/tex]
- [tex]\( P(x) \)[/tex] = [tex]\( \left\{\frac{3}{17}, \frac{5}{17}, \frac{6}{17}, \frac{2}{17}, \frac{1}{17} \right\} \)[/tex]
We calculate the mean as follows:
[tex]\[ \mu = (0 \cdot \frac{3}{17}) + (1 \cdot \frac{5}{17}) + (2 \cdot \frac{6}{17}) + (3 \cdot \frac{2}{17}) + (4 \cdot \frac{1}{17}) \][/tex]
[tex]\[ \mu = 0 + \left(\frac{5}{17}\right) + \left(\frac{12}{17}\right) + \left(\frac{6}{17}\right) + \left(\frac{4}{17}\right) \][/tex]
[tex]\[ \mu = \frac{5 + 12 + 6 + 4}{17} = \frac{27}{17} \approx 1.59 \][/tex]
So, the mean is:
[tex]\[ \text{mean} = 1.59 \][/tex]
### Standard Deviation
The standard deviation is the square root of the variance. The variance ([tex]\( \sigma^2 \)[/tex]) is calculated using the formula:
[tex]\[ \sigma^2 = \sum (P(x_i) \cdot (x_i - \mu)^2) \][/tex]
To find the variance, we need the values of [tex]\( (x_i - \mu)^2 \)[/tex]:
[tex]\[ (0 - 1.59)^2 = 2.5281 \][/tex]
[tex]\[ (1 - 1.59)^2 = 0.3481 \][/tex]
[tex]\[ (2 - 1.59)^2 = 0.1681 \][/tex]
[tex]\[ (3 - 1.59)^2 = 2.0161 \][/tex]
[tex]\[ (4 - 1.59)^2 = 5.8561 \][/tex]
Now we calculate the variance:
[tex]\[ \sigma^2 = ( \frac{3}{17} \cdot 2.5281 ) + ( \frac{5}{17} \cdot 0.3481 ) + ( \frac{6}{17} \cdot 0.1681 ) + ( \frac{2}{17} \cdot 2.0161 ) + ( \frac{1}{17} \cdot 5.8561 ) \][/tex]
[tex]\[ \sigma^2 = \left( \frac{7.5843}{17} \right) + \left( \frac{1.7405}{17} \right) + \left( \frac{1.0086}{17} \right) + \left( \frac{4.0322}{17} \right) + \left( \frac{5.8561}{17} \right) \][/tex]
[tex]\[ \sigma^2 = \frac{7.5843 + 1.7405 + 1.0086 + 4.0322 + 5.8561}{17} = \frac{20.2217}{17} \approx 1.189 \][/tex]
The standard deviation is the square root of the variance:
[tex]\[ \sigma = \sqrt{1.189} \approx 1.09 \][/tex]
So, the standard deviation is:
[tex]\[ \text{standard deviation} = 1.09 \][/tex]
### Answers
a) Mean [tex]\( = 1.59 \)[/tex]
b) Standard Deviation [tex]\( = 1.09 \)[/tex]
We appreciate your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Thank you for visiting IDNLearn.com. We’re here to provide accurate and reliable answers, so visit us again soon.
