A die shows up numbers one to six with probabilities 0.1, 0.1, 0.1, 0.1, 0.2, 0.4, respectively.  Roll the die four times independently. It is known that for i=1,2,3 4, the number i is not observed in the i-th roll.  What is the probability of getting the number 5 once and the number 6 once.

It is known that 1% of chips from a product line are defective.  A professor who bought 1000 chips found out that 30 of them are defective.   Now randomly select 200 from the 1000 chips with replacement. What is the probability that 5 are defective?

Suppose we have 10 observations from an exponential distribution with mean 10 and standard deviation 10.  Denote the sample mean and sample variance as 

Two discrete random variables X and Y have the joint probability function f(x,y): f(x,y) x=0 1 2 y=0 0.10 0.20 0.10 2 0 0.10 0.10 4 0.20 0 0.20               Determine whether X and Y are independent ,  True or False    

An engineer has 20 red marbles and 30 purple marbles. He mixes them, randomly put them in two boxes with equal numbers of marbles.  Now he randomly selects a box and takes out two marbles without replacement.  What is the probability of getting one red marble?

Roll a pair of balance dice five times. Find the probability of getting a sum of 11 once and a matching pair once.

A random sample of size 64 is from an exponential distribution with parameter =1.2.  See page 195 for the definition of the exponential distribution.    Let 

The following observations are the numbers of customers in a service center in ten days.                9, 9, 8, 13, 19, 5, 11, 12, 10, 8 Find the 10% trimmed mean.

Two independent samples of sizes are randomly selected from two normal populations with unknow variances,   The sample means and the sample variances are as follows:  

In a state, wildfires occur 4 times every year on average. Assume the number of wildfires follows from Poisson distribution.  Find the probability that from 13 to 15 wildfires can occur from January 1, 2029 to December 31, 2032?