Question

If 15 observations are taken independently from a chi-square distribution with fifteen degrees of freedom, find the probability that at most ten of the 15 observations are less than 10. Show your work.

Answer 1

Let be 15 observations from chi square distribution with 15 degrees of freedom.

Let be the probability that an observation less than 10.

Then

Let .

Define success as if an observations from chi square distribution with 15 degrees of freedom is less than 10

Then are independent and identically distributed Bernoulli random variables with probability of success

Define . Then Y is the total number of observations less than 10, and it has Binomial distribution with parameters n=15 and .

So the probability mass function of Y is given by

Now the probability that at most ten of the 15 observations are less than 10 is same as