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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]
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