# Drug Probability Problem

• JaysFan31
In summary, a university probability class is studying multivariate distributions and a problem involving a new drug for arthritis. The proportion p of patients who respond favorably to the drug is a random variable with a probability density function given by f(p)={12*(p^(2))*(1-p), when p is between 0 and 1 inclusive. The number of patients showing a favorable response, denoted by Y, varies depending on the batch of the drug and the number of patients injected. The unconditional probability distribution of Y for general n can be found by integrating the function from 0 to 1. E(Y) for n=2 can also be found by integrating the function from 0 to 1.
JaysFan31
I'm in a university probability class studying multivariate distributions and have a problem I'm stuck on.

Here goes:
In a clinical study of a new drug formulated to reduce the effects of arthritis, researchers found that the proportion p of patients who respond favourably to the drug is a random variable that varies from batch to batch of the drug. Assume that p has a probability density function given by
f(p)={12*(p^(2))*(1-p), whenever p is between 0 and 1 inclusive
{0, whenever p is elsewhere.
Suppose that n patients are injected with portions of the drug taken from the same batch. Let Y denote the number showing a favourable response.
(A) Find the unconditional probability distribution of Y for general n.
(B) Find E(Y) for n = 2.

I'm confused because there's no Y1 and Y2. Every problem I've done has Y1 and Y2. How do you find the unconditional probability distribution for just Y in this case? I would love any help. Just a suggestion needed.

The proportion of patients who respond favorably to the drug is, by DEFINITION, the number of patients who responded favorably divided by the total number of patients injected with the drug.
If n patients are injected with the drug and the proportion who respond favorably is f(p) then the NUMBER who respond favorably is
nf(p), of course!

Thanks for the response. I need to integrate the function from 0 to 1 I presume? What's E(Y) though?

Since Y= nf(p), E(Y) is the integral of that, from 0 to 1.

## 1. What is the "Drug Probability Problem"?

The "Drug Probability Problem" is a theoretical question that asks: If a drug has a certain probability of success for each patient, what is the probability that at least one patient will experience success after a given number of trials?

## 2. Why is the "Drug Probability Problem" important?

The "Drug Probability Problem" is important because it helps researchers and scientists understand the likelihood of success for a new drug and can inform decisions about its effectiveness and potential impact on patients.

## 3. How is the "Drug Probability Problem" solved?

The "Drug Probability Problem" can be solved using the binomial distribution formula, which takes into account the probability of success, number of trials, and number of desired successes. This formula can be used to calculate the probability of at least one success in a given number of trials.

## 4. What factors can affect the solution to the "Drug Probability Problem"?

The solution to the "Drug Probability Problem" can be affected by several factors, including the probability of success for the drug, the number of trials, and the number of desired successes. Additionally, external factors such as patient characteristics and study design can also impact the outcome.

## 5. How can the "Drug Probability Problem" be applied in real-world scenarios?

The "Drug Probability Problem" can be applied in real-world scenarios to help researchers and scientists make informed decisions about the effectiveness of a new drug. It can also be used to estimate the number of patients needed in a clinical trial to achieve a desired level of success, and to determine the probability of success for a new drug in a given population.

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