Get expert advice and community support on IDNLearn.com. Join our knowledgeable community to find the answers you need for any topic or issue.
Sagot :
Sure, let's work through this step by step for each part of the question.
### (a) Compute [tex]\( f(11) \)[/tex]
The probability mass function for a binomial distribution is given by:
[tex]\[ f(k) = P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
For [tex]\( n = 20 \)[/tex], [tex]\( p = 0.70 \)[/tex], and [tex]\( k = 11 \)[/tex]:
[tex]\[ f(11) \approx 0.0654 \][/tex]
### (b) Compute [tex]\( f(16) \)[/tex]
Using the same probability mass function for a binomial distribution:
For [tex]\( n = 20 \)[/tex], [tex]\( p = 0.70 \)[/tex], and [tex]\( k = 16 \)[/tex]:
[tex]\[ f(16) \approx 0.1304 \][/tex]
### (c) Compute [tex]\( P(x \geq 16) \)[/tex]
The probability of [tex]\( x \)[/tex] being greater than or equal to 16 is the sum of individual probabilities from 16 to 20. This is often easier calculated using the survival function:
[tex]\[ P(x \geq 16) \approx 0.2375 \][/tex]
### (d) Compute [tex]\( P(x \leq 15) \)[/tex]
The probability of [tex]\( x \)[/tex] being less than or equal to 15 is the cumulative distribution function (CDF) evaluated at 15:
[tex]\[ P(x \leq 15) \approx 0.7625 \][/tex]
### (e) Compute [tex]\( E(x) \)[/tex]
The expected value [tex]\( E(x) \)[/tex] for a binomial distribution is given by:
[tex]\[ E(x) = n \cdot p \][/tex]
For [tex]\( n = 20 \)[/tex] and [tex]\( p = 0.70 \)[/tex]:
[tex]\[ E(x) = 20 \cdot 0.70 = 14.0 \][/tex]
### (f) Compute [tex]\( \operatorname{Var}(x) \)[/tex] and [tex]\( \sigma \)[/tex]
The variance [tex]\( \operatorname{Var}(x) \)[/tex] for a binomial distribution is:
[tex]\[ \operatorname{Var}(x) = n \cdot p \cdot (1 - p) \][/tex]
For [tex]\( n = 20 \)[/tex] and [tex]\( p = 0.70 \)[/tex]:
[tex]\[ \operatorname{Var}(x) = 20 \cdot 0.70 \cdot 0.30 = 4.2000 \][/tex]
The standard deviation [tex]\( \sigma \)[/tex] is the square root of the variance:
[tex]\[ \sigma = \sqrt{\operatorname{Var}(x)} = \sqrt{4.2000} \approx 2.0494 \][/tex]
### Summary:
- [tex]\( f(11) \approx 0.0654 \)[/tex]
- [tex]\( f(16) \approx 0.1304 \)[/tex]
- [tex]\( P(x \geq 16) \approx 0.2375 \)[/tex]
- [tex]\( P(x \leq 15) \approx 0.7625 \)[/tex]
- [tex]\( E(x) = 14.0 \)[/tex]
- [tex]\( \operatorname{Var}(x) = 4.2000 \)[/tex]
- [tex]\( \sigma \approx 2.0494 \)[/tex]
### (a) Compute [tex]\( f(11) \)[/tex]
The probability mass function for a binomial distribution is given by:
[tex]\[ f(k) = P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \][/tex]
For [tex]\( n = 20 \)[/tex], [tex]\( p = 0.70 \)[/tex], and [tex]\( k = 11 \)[/tex]:
[tex]\[ f(11) \approx 0.0654 \][/tex]
### (b) Compute [tex]\( f(16) \)[/tex]
Using the same probability mass function for a binomial distribution:
For [tex]\( n = 20 \)[/tex], [tex]\( p = 0.70 \)[/tex], and [tex]\( k = 16 \)[/tex]:
[tex]\[ f(16) \approx 0.1304 \][/tex]
### (c) Compute [tex]\( P(x \geq 16) \)[/tex]
The probability of [tex]\( x \)[/tex] being greater than or equal to 16 is the sum of individual probabilities from 16 to 20. This is often easier calculated using the survival function:
[tex]\[ P(x \geq 16) \approx 0.2375 \][/tex]
### (d) Compute [tex]\( P(x \leq 15) \)[/tex]
The probability of [tex]\( x \)[/tex] being less than or equal to 15 is the cumulative distribution function (CDF) evaluated at 15:
[tex]\[ P(x \leq 15) \approx 0.7625 \][/tex]
### (e) Compute [tex]\( E(x) \)[/tex]
The expected value [tex]\( E(x) \)[/tex] for a binomial distribution is given by:
[tex]\[ E(x) = n \cdot p \][/tex]
For [tex]\( n = 20 \)[/tex] and [tex]\( p = 0.70 \)[/tex]:
[tex]\[ E(x) = 20 \cdot 0.70 = 14.0 \][/tex]
### (f) Compute [tex]\( \operatorname{Var}(x) \)[/tex] and [tex]\( \sigma \)[/tex]
The variance [tex]\( \operatorname{Var}(x) \)[/tex] for a binomial distribution is:
[tex]\[ \operatorname{Var}(x) = n \cdot p \cdot (1 - p) \][/tex]
For [tex]\( n = 20 \)[/tex] and [tex]\( p = 0.70 \)[/tex]:
[tex]\[ \operatorname{Var}(x) = 20 \cdot 0.70 \cdot 0.30 = 4.2000 \][/tex]
The standard deviation [tex]\( \sigma \)[/tex] is the square root of the variance:
[tex]\[ \sigma = \sqrt{\operatorname{Var}(x)} = \sqrt{4.2000} \approx 2.0494 \][/tex]
### Summary:
- [tex]\( f(11) \approx 0.0654 \)[/tex]
- [tex]\( f(16) \approx 0.1304 \)[/tex]
- [tex]\( P(x \geq 16) \approx 0.2375 \)[/tex]
- [tex]\( P(x \leq 15) \approx 0.7625 \)[/tex]
- [tex]\( E(x) = 14.0 \)[/tex]
- [tex]\( \operatorname{Var}(x) = 4.2000 \)[/tex]
- [tex]\( \sigma \approx 2.0494 \)[/tex]
We greatly appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for visiting IDNLearn.com. We’re here to provide accurate and reliable answers, so visit us again soon.
