The statement some dogs are brown has 16 letters. Choose one of these 16 letters uniformly at random (i.e., with equal probability). Let X denote the length of the word containing the chosen letter. Determine the possible values and probability mass function of X.

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