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

Answer:

y                  3                   4                 5

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P(Y)           0.1875          0.5             0.3125

Step-by-step 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:

$P(SOME) = \frac{4}{16}= \frac{1}{4}=0.25$

And similar for the other words we have:

$P(DOGS) = \frac{4}{16}= \frac{1}{4}=0.25$

$P(ARE) = \frac{3}{16}= 0.1875$

$P(BROWN)=\frac{5}{16}= 0.3125$

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