The length of stay at a hospital emergency department
is the sum of the waiting and service times. Let X denote the
proportion of time spent waiting and assume a beta distribution
with α =10 and β =1. Determine the following:
(a) P(X > 0.9) (b) P(X < 0.5) (c) Mean and variance
To determine the probabilities and statistics related to the proportion of time spent waiting at a hospital emergency department, which is denoted by X and assumed to follow a beta distribution with α = 10 and β = 1, we can use the properties of the beta distribution.
The probability density function (PDF) of a beta distribution is given by:
f(x) = (x^(α-1) * (1-x)^(β-1)) / B(α, β)
Where x is the proportion of time spent waiting, α is the shape parameter, β is the shape parameter, and B(α, β) is the beta function.
(a) P(X > 0.9):
To find this probability, we need to calculate the cumulative distribution function (CDF) and subtract it from 1.
P(X > 0.9) = 1 - P(X ≤ 0.9) = 1 - F(0.9)
To calculate F(0.9), we integrate the PDF from 0 to 0.9. However, since the beta distribution has a complex integral, we usually use statistical software or online calculators to find the CDF directly. Alternatively, if you have access to a statistical software like R or Python, you can use the corresponding functions to calculate it. For example, in R, you can use the pbeta() function:
P(X > 0.9) = 1 - pbeta(0.9, 10, 1)
(b) P(X < 0.5):
Similarly, to calculate this probability, we can directly use the CDF:
P(X < 0.5) = F(0.5)
Again, you can use statistical software or online calculators to compute the CDF. In R, the pbeta() function can be used:
P(X < 0.5) = pbeta(0.5, 10, 1)
(c) Mean and variance:
The mean (μ) and variance (σ^2) of a beta distribution with parameters α and β can be calculated using the following formulas:
μ = α / (α + β)
σ^2 = (α * β) / [(α + β)^2 * (α + β + 1)]
For the given distribution with α = 10 and β = 1:
μ = 10 / (10 + 1) = 10 / 11
σ^2 = (10 * 1) / [(10 + 1)^2 * (10 + 1 + 1)] = 10 / (11^3)
Therefore, the mean of X is 10/11 and the variance is 10/(11^3).