Answer:
Rx = {5,4,3}
The probability mass function can be formulated like this:
Px (3) = p (X = 3) = 3/16
Px (4) = p (X = 4) = 8/16
Px (5) = p (X = 5) = 5/16
Step-by-step explanation:
We have that the probability of choosing a letter (1) at random from the sentence would be 1/16.
We must first finish the length of each word:
some = 4 letters
dogs = 4 letters
are = 3 letters
brown = 5 letters
Therefore, the possible values, that is, the range would be:
Rx = {5,4,3}
since the range has finite set, then the random variable X is discrete random variable
now to find the probability mass function of X and it can be defined as:
Px (X) = P (X = x); where x = 3, 4, 5
Px (3) = p (X = 3)
It will be equal sum of probalities of all the letters that can form the three letter word "are" by letters A, R, E
Probability of getting length 3 when a letter is drawm from the some dogs are brown:
1/16 + 1/16 + 1/16 = 3/16
Px (4) = p (X = 4)
It will be equal sum of probalities of all the letters that can form the four letter word "some" and "dogs" by letters S, O, M, E, D, O, G, S
Probability of getting length 4 when a letter is drawm from the some dogs are brown:
1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 +1/16 + 1/16 = 8/16
Px (5) = p (X = 5)
It will be equal sum of probalities of all the letters that can form the five letter word "brown" by letters B, R, O, W, N
Probability of getting length 5 when a letter is drawm from the some dogs are brown:
1/16 + 1/16 + 1/16 + 1/16 + 1/16 = 5/16
X 3 4 5
p(X) 3/16 8/16 5/16
