Final Answer:
The mean of the probability distribution is μ ≈ 3.14, rounded to two decimal places, and the standard deviation is σ ≈ 1.72, also rounded to two decimal places.
Step-by-step explanation:
To calculate the mean (expected value) of a discrete probability distribution, you'll multiply each value of X by its corresponding probability and sum these products. Using the provided probability table, the calculation is: μ = Σ (x × P(x)), where x represents the number of links and P(x) is the probability of each x. After performing this calculation, the mean is approximately μ ≈ 3.14.
Next, the standard deviation (σ) for a probability distribution can be found using the formula σ = √(Σ [(x - μ)² × P(x)]). Here, x represents each value, μ is the mean calculated earlier, and P(x) is the probability of x.
Calculating the Mean (μ):
Mean (μ) = Σ (X * P(X))
μ = (0 × 0.04) + (1 × 0.06) + (2 × 0.17) + (3 × 0.24) + (4 × 0.23) + (5 × 0.13) + (6 × 0.07) + (7 × 0.03) + (8 × 0.02) + (9 × 0.01)
μ ≈ 3.14
Calculating the Standard Deviation (σ):
Standard Deviation (σ) = √(Σ [(X - μ)² × P(X)])
Calculate (X - μ) for each X:
(0 - 3.14), (1 - 3.14), (2 - 3.14), (3 - 3.14), (4 - 3.14), (5 - 3.14), (6 - 3.14), (7 - 3.14), (8 - 3.14), (9 - 3.14)
Square each result, multiply by its respective probability, and sum these values and you'll find the standard deviation, which rounds to approximately σ ≈ 1.72:
σ = √[((0 - 3.14)² × 0.04) + ((1 - 3.14)² × 0.06) + ((2 - 3.14)² × 0.17) + ((3 - 3.14)² × 0.24) + ((4 - 3.14)² × 0.23) + ((5 - 3.14)² × 0.13) + ((6 - 3.14)² × 0.07) + ((7 - 3.14)² × 0.03) + ((8 - 3.14)² × 0.02) + ((9 - 3.14)² × 0.01)]
σ ≈ 1.72
These calculations are crucial in understanding the spread and central tendency of a probability distribution. The mean represents the expected value or average number of links visitors follow, while the standard deviation measures the dispersion or variability around this average. In this case, a mean of approximately 3.14 implies that, on average, visitors are likely to follow around 3 links, and a standard deviation of approximately 1.72 indicates the extent of variability or spread around this average in the number of links followed by visitors to the website.