Question

question 1 (1 point)
evaluate dbinom(3,5,0.40), give your answer to round to nearest six decimal places.
your answer:
answer

question 2 (1 point)
evaluate pbinom(2,9,0.45), give your answer to round to nearest six decimal places.
your answer:
answer

question 3 (1 point)
evaluate dpois(10,4.65), give your answer to round to nearest six decimal places.
your answer:
answer

Explanation:

Understand the R functions

In R, statistical distributions have specific prefix functions:

• dbinom(x, size, prob) calculates the probability mass function (PMF) of a binomial distribution: \(P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}\).
• pbinom(q, size, prob) calculates the cumulative distribution function (CDF) of a binomial distribution: \(P(X \le q) = \sum_{i=0}^{q} \binom{n}{i} p^i (1-p)^{n-i}\).
• dpois(x, lambda) calculates the probability mass function (PMF) of a Poisson distribution: \(P(X = x) = \frac{e^{-\lambda} \lambda^x}{x!}\).

Evaluate Question 1

Using the Discrete Probability Distribution knowledge point:
We need to evaluate dbinom(3, 5, 0.40), which represents \(P(X = 3)\) for a binomial distribution with \(n = 5\) and \(p = 0.40\).

$$ P(X = 3) = \binom{5}{3} (0.40)^3 (1 - 0.40)^{5 - 3} $$

$$ P(X = 3) = 10 \times 0.064 \times 0.36 = 0.230400 $$

Evaluate Question 2

Using the Discrete Probability Distribution knowledge point:
We need to evaluate pbinom(2, 9, 0.45), which represents \(P(X \le 2)\) for a binomial distribution with \(n = 9\) and \(p = 0.45\).

$$ P(X \le 2) = P(X = 0) + P(X = 1) + P(X = 2) $$

$$ P(X = 0) = \binom{9}{0} (0.45)^0 (0.55)^9 \approx 0.00460051 $$

$$ P(X = 1) = \binom{9}{1} (0.45)^1 (0.55)^8 \approx 0.03389148 $$

$$ P(X = 2) = \binom{9}{2} (0.45)^2 (0.55)^7 \approx 0.11101030 $$

$$ P(X \le 2) \approx 0.00460051 + 0.03389148 + 0.11101030 = 0.14950229 \approx 0.149502 $$

Evaluate Question 3

Using the Poisson Distribution knowledge point:
We need to evaluate dpois(10, 4.65), which represents \(P(X = 10)\) for a Poisson distribution with mean \(\lambda = 4.65\).

$$ P(X = 10) = \frac{e^{-4.65} (4.65)^{10}}{10!} $$

$$ e^{-4.65} \approx 0.00956158 $$

$$ (4.65)^{10} \approx 4402685.25 $$

$$ 10! = 3628800 $$

$$ P(X = 10) \approx \frac{0.00956158 \times 4402685.25}{3628800} \approx 0.01159981 \approx 0.011600 $$

Answer:

Question 1

Evaluate dbinom(3,5,0.40), Give your answer to round to nearest six decimal places.

Your Answer:
<blank>0.230400</blank>

Question 2

Evaluate pbinom(2,9,0.45), Give your answer to round to nearest six decimal places.

Your Answer:
<blank>0.149502</blank>

Question 3

Evaluate dpois(10,4.65), Give your answer to round to nearest six decimal places.

Your Answer:
<blank>0.011600</blank>