Respuesta :
Answer:
(a) Probabilities:
- \( P(X=0) = 0.071 \)
- \( P(X=1) = 0.286 \)
- \( P(X=2) = 0.357 \)
- \( P(X=3) = 0.107 \)
- \( P(X=4) = 0.107 \)
- \( P(X=5) = 0.035 \)
- \( P(X=6) = 0.035 \)
(b) \( P(X=2) = 0.357 \)
(c) \( P(X \geq 4) = 0.107 \)
(d) The mean of X is 2.124.
(e) The standard deviation of X is approximately 1.682.
Step-by-step explanation:
(d) To find the mean of X, we'll use the formula for the expected value of a discrete random variable:
\[E(X) = \sum_{i=1}^{k} x_i \cdot P(X=x_i)\]
where \(x_i\) is the value of the random variable and \(P(X=x_i)\) is the corresponding probability.
Using the given data:
\[E(X) = (0 \cdot 0.071) + (1 \cdot 0.286) + (2 \cdot 0.357) + (3 \cdot 0.107) + (4 \cdot 0.107) + (5 \cdot 0.035) + (6 \cdot 0.035)\]
\[E(X) = 0 + 0.286 + 0.714 + 0.321 + 0.428 + 0.175 + 0.21\]
\[E(X) = 2.124\]
So, the mean of X is 2.124.
(e) To find the standard deviation of X, we'll use the formula:
\[ \sigma = \sqrt{\sum_{i=1}^{n} (x_i - \mu)^2 \cdot P(X=x_i)} \]
where \(x_i\) is the value of the random variable, \(\mu\) is the mean, and \(P(X=x_i)\) is the corresponding probability.
We already know \(\mu = 2.124\). Now, we'll calculate \(\sigma\):
\[ \sigma = \sqrt{(0-2.124)^2 \cdot 0.071 + (1-2.124)^2 \cdot 0.286 + (2-2.124)^2 \cdot 0.357 + (3-2.124)^2 \cdot 0.107 + (4-2.124)^2 \cdot 0.107 + (5-2.124)^2 \cdot 0.035 + (6-2.124)^2 \cdot 0.035}\]
\[ \sigma = \sqrt{(4.516 \cdot 0.071) + (1.903 \cdot 0.286) + (0.015 \cdot 0.357) + (2.561 \cdot 0.107) + (5.409 \cdot 0.107) + (14.196 \cdot 0.035) + (17.536 \cdot 0.035)}\]
\[ \sigma = \sqrt{0.320936 + 0.544858 + 0.005355 + 0.273127 + 0.578943 + 0.49786 + 0.61456}\]
\[ \sigma ≈ \sqrt{2.835639} ≈ 1.682\]
So, the standard deviation of X is approximately 1.682.
