# 3.A recent article in the Journal of Urban Chicken Farmers reported the fraction of chickens thatexperience severe, moderate, or minor side effects from a new diet are 0.07, 0.15, and 0.78,respectively. Assuming that chickens respond independently, if 18 chickens were tested on the new diet,what is theProbability that 2, 4, and 12 chickens will suffer severe, moderate, or minor side effects,respectively?Probability that no chickens will suffer severe side effects?Mean number of chickens that will suffer severe side effects?Variance in the number of chickens that will suffer severe side effects?Conditional mean of the number of chickens that suffer severe side effects given that 17suffer minor side effects.a.b.C.d.e.

Question
1 views
check_circle

Step 1

The distribution:

Denote X1, X­2 and X3 as the numbers of chickens out of the n = 18 tested chickens, which experience severe, moderate and minor side effects respectively. Here, X1 = 2, X­2 = 4, X3 = 12, X1 + X­2 + X3 = 18.

The probabilities for having severe, moderate and minor side effects are respectively p1 = 0.07, p­2 = 0.15, p3 = 0.78, p1 + p2 + p3 = 1.

The reactions of the chickens are independent of each other.

Thus, the number of chickens with the three types of side effects has a multinomial distribution.

Part (a):

The probability that 2, 4 and 12 chickens will suffer severe, moderate and minor side effects respectively is:

Step 2

Part (b):

For a multinomial distribution, each Xi marginally follows the binomial distribution with parameters n and pi, for i = 1, 2, …, k.

Thus, the probability that no chickens suffer severe side effects is:

Step 3

Part (c):

The binomial mean with parameters n and p is np.

Thus, the mean number of chic...

### Want to see the full answer?

See Solution

#### Want to see this answer and more?

Solutions are written by subject experts who are available 24/7. Questions are typically answered within 1 hour.*

See Solution
*Response times may vary by subject and question.
Tagged in