Binomial distribution is a very important and common distribution in statistics. A binomial variable X is always COUNT of something. But not all count variables have binomial distributions. There are 4 binomial settings to justify whether a variable X is a Binomial variable or not.

a. There are a fixed number n of observations. b. The n observations are all independent. c. Each observation falls into just two categories. We define one of them as 'Success', and the other 'Failure'. d. The probability of success for a single observation (p) is the same for each observation. Keep these in mind, and tell whether the variable X is a binomial variable or not. It has to satisfy all 4 settings to be binomial variable. If X is a binomial variable, write in the form X- Bin (n, p), with n and p values specified. If X is not a binomial variable, which of the 4 settings are violated? (1) (2.5pts)Research shows that around 90% of the chicken eggs will hatch at 100F. X is the number of hatched eggs from an incubator set as 100 F in a farm. (2) (2.5pts)Research shows that around 90% of the chicken eggs will hatch at 100 F, but only 80% at 95 F. In a farm, there are two incubators, and each have 20 eggs to hatch. One incubator is set at 100 F and the other is set at 95 F. X is the total number of hatched eggs from both incubators. (3) (2.5pts)Research shows that chicken eggs and duck eggs hatch rate at 100F are both 90%. 10 chicken eggs and 10 duck eggs are put in an incubator controlled at 100 F. X is the number of hatched eggs from the incubator. (4) (2.5pts)When an opinion poll calls residential telephone numbers at random, only 20% of the phone calls reach a live person. X is the number of calls that reach a live person out of 20 dialings from the random digit dialing machine.