Question

Statistics W2A1:

Part 1

Provide an example of a probability distribution that has four distinct outcomes. In this example, please identify the outcome and the probability of that outcome occurring.

Part 2

A die is rolled 20 times and the number of “fives” that occur is reported as being the random variable. Is this a binomial random variable? Why or why not

Please answer thoroughly!

Answer 1

Ans.

Part1.

Suppose we have a fair die of four faces with numbers on the faces 1, 2, 3, 4.

When throughing such die, then the sample space is given by

S = ( 1, 2, 3, 4 )

Let X be the number on the upper face of the die, then it's probability distribution is given by

P(X = i) = 1/4 = 0.25, for i = 1,2,3,4

Which is pmf of the discrete uniform distribution since all the outcomes are equally likely.

Part2.

Suppose we have a fair die of six faces with numbers on the faces are 1,2,3,4,5,6.

When throughing such die then the sample space is given by

S = (1,2,3,4,5,6)

Since this experiment is repeated 20 times, each trail is independently ferfom with constant probability of success

p = p(five) = 1/6

Let X be the number of five

Since we have n = 20 independent trials each with constant probability of success p = 1/6 and the probability of failure

q = 1 - p = 5/6

Then the random variable Xhas binomial distribution with parameters n= 20 and p = 1/6.