Suppose the fraction of a cohort of aids patients that survives a time t after aids diagnosis is given by S(t) = exp(-kt). show that the average survival time Taver after aids diagnosis for a member of this cohort is given by Taver= 1/k. show every steps for getting Taver = 1/k
note that
∫[0,∞] e-kx
= -1/k e-kx [0,∞]
= 1/k
To derive the average survival time Taver for a member of the AIDS patient cohort, we need to calculate the integral of the survival function S(t) over the entire range of time. Here's a step-by-step approach for solving the problem:
Step 1: Given the survival function S(t) = exp(-kt), where t represents time and k is a constant.
Step 2: To find the average survival time Taver, we need to calculate the integral of S(t) over time. Mathematically, this can be written as follows:
Taver = ∫(0 to ∞) t * S(t) dt
Step 3: Substitute the given expression for S(t) into the integral:
Taver = ∫(0 to ∞) t * exp(-kt) dt
Step 4: Integrate the equation using integration by parts:
Let u = t and dv = exp(-kt) dt
du = dt and v = -exp(-kt)/k
Using the integration by parts formula:
∫u * dv = uv - ∫v * du
Taver = [(-t * exp(-kt))/k]∣(0 to ∞) - ∫(0 to ∞) (-exp(-kt)/k) dt
Step 5: Evaluate the definite integral on the right-hand side of the equation:
Taver = [(-t * exp(-kt))/k]∣(0 to ∞) + [exp(-kt)/k^2]∣(0 to ∞)
Taver = [(0 * exp(-k(0)))/k - (-∞ * exp(-k(∞)))/k] + [exp(-k(∞))/k^2 - exp(-k(0))/k^2]
Since exp(-k(∞)) = 0, the second term becomes:
Taver = [0 - 0] + [0 - exp(-k(0))/k^2]
Taver = -exp(-k(0))/k^2
Step 6: Simplify the expression using the property that exp(0) = 1:
Taver = -1/k^2
Step 7: Multiply both the numerator and denominator by -1:
Taver = 1/(-k^2)
Step 8: Take the reciprocal of the equation:
Taver = -1/(k^2)
Finally, flipping the equation to put the constant term on the denominator:
Taver = 1/k^2
Step 9: Take the square root of the equation:
Taver = 1/k
Therefore, the average survival time Taver for a member of the AIDS patient cohort is given by Taver = 1/k.