Question

6. News articles that link to related stories are widely used in Web marketing. With a large number of daily visitors to a Web page, we model the proportion of daily visitors who click on a link to a related story as approximately a continuous random variable with a beta distribution. The parameters are α=6 and β=1. a. What is the mean and standard deviation of the proportion of visitors who click? b. What is the probability a proportion exceeds 0.58 ? c. What proportion is exceeded with probability 0.33 ? d. If 550 visitors view the page, what is the expected number of visitors who click on a link to a related story?

Answer 1

a. The mean of the proportion of visitors who click is 6/7, and the standard deviation is 1/7. b. The probability that a proportion exceeds 0.58 can be calculated using the CDF of the beta distribution. c. The proportion that is exceeded with probability 0.33 can be calculated using the ICDF of the beta distribution. d. The expected number of visitors who click can be calculated by multiplying the total number of visitors by the mean proportion of visitors who click.

Step-by-step explanation:

a. Mean: The mean of a beta distribution with parameters α and β is given by the formula μ = α / (α + β). In this case, the mean is 6 / (6 + 1) = 6/7.

Standard Deviation: The standard deviation of a beta distribution with parameters α and β is given by the formula σ = sqrt((α * β) / ((α + β)^2 * (α + β + 1))). In this case, the standard deviation is sqrt((6 * 1) / ((6 + 1)^2 * (6 + 1 + 1))) = sqrt(6/294) = sqrt(1/49) = 1/7.

b. Probability of exceeding 0.58: To find the probability that a proportion exceeds 0.58, we can use the cumulative distribution function (CDF) of the beta distribution. In this case, the probability is 1 - CDF(0.58), which can be calculated using a software or calculator that has the beta distribution function.

c. Proportion exceeded with probability 0.33: To find the proportion that is exceeded with probability 0.33, we can use the inverse cumulative distribution function (ICDF) of the beta distribution. In this case, the proportion is ICDF(0.33), which can be calculated using a software or calculator that has the beta distribution function.

d. Expected number of visitors who click: The expected number of visitors who click on a link to a related story can be calculated by multiplying the total number of visitors (550) by the mean proportion of visitors who click. In this case, the expected number is 550 * (6/7) = 471.43 (rounded to the nearest whole number).

Answer 2

1. a. The mean of the proportion of visitors who click X is 0.86, and the standard deviation is 0.09.

b. The probability that the proportion exceeds 0.58 is approximately0.0046.

c. The proportion that is exceeded with a probability of 0.33 is approximately 0.81.

d. The expected number of visitors who click on a link to a related story, given that 550 visitors view the page, is 473.

Step-by-step explanation:

a. The mean
`(\(\mu\))` and standard deviation
`(\(\sigma\))`of a beta distribution with parameters
`\(\alpha\)` and
`\(\beta\)`are given by
`\(\mu = (\alpha)/(\alpha + \beta)\)`and
`\(\sigma = \sqrt{(\alpha\beta)/((\alpha+\beta)^2(\alpha+\beta+1))}\).` Substituting
`\(\alpha = 6\)` and
`\(\beta = 1\)`, we find
`\(\mu = (6)/(7) \approx 0.86\) and \(\sigma \approx 0.09\).`

b. To find the probability that the proportion exceeds 0.58, we use the cumulative distribution function (CDF) of the beta distribution.
`\(P(X > 0.58) = 1 - P(X \leq 0.58)\)`. Using the beta distribution function, we find this probability to be approximately \(0.0046\).

c. To determine the proportion that is exceeded with a probability of 0.33, we use the inverse of the cumulative distribution function (CDF).
`\(P(X > x) = 0.33\)`. Solving for x, we find
`\(x \approx 0.81\)`.

d. The expected number of visitors who click
`(\(E(X)\))` given 550 visitors is calculated as
`\(E(X) = \mu * \text{Total Visitors} = 0.86 * 550 = 473\).`