Question

The statement SOME DOGS ARE BROWN has 16 letters. Choose one of the 16 letters uniformly at random. Let X denote the length of the word containing the chosen letter. Determine the possible values and probability mass function of X.

Answer 1

y 3 4 5

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P(Y) 0.1875 0.5 0.3125

Explanation:

For this case we have the following statement : SOME DOGS ARE BROWN

And if we count the number of letters in the statement we have 16 possible letters. We can define the following random variable

Y= Length of word containing a selected letter.

For example the word DOGS have 4 of the total 16 letters, so we can define the probability using empirical approximation for each word like this:

`P(SOME) = (4)/(16)= (1)/(4)=0.25`

And similar for the other words we have:

`P(DOGS) = (4)/(16)= (1)/(4)=0.25`

`P(ARE) = (3)/(16)= 0.1875`

`P(BROWN)=(5)/(16)= 0.3125`

And we have the following distribution for the words in the statement with the random variable Y defined previously:

Word SOME DOGS ARE BROWN

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y 4 4 3 5

P(Word) 0.25 0.25 0.1875 0.3125

And as we can see the possible values for Y are 3,4 and 5, so then we can define the probability distribution for Y like this:

y 3 4 5

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P(Y) 0.1875 0.5 0.3125