Question

Let X denote the number of vowels from this sentence. Find the expected value and standard deviation of the random variable X.

Answer 1

The expected value and standard deviation of the random variable X:

Explanation:

• There are 12 words in the following statement.

• It is clear that X can take values 0,1,2,3,4,5.Then,we have,

• P (X = 0) = P (Selecting a word containing no vowel) = P (Selecting X) =  1/12.
• P (X = 1) = P (Selecting a word containing one vowel)
• P (X = 1) = P (Selecting a word from the words 'The', 'Find', 'and', of 'The') = 5/12.
• P(X=2) = 2/12, P(X= 3) = 2/12, P(X= 4) =1/12, P(X= 5) =1/12.    
• Therefore, E(X) = 2
• E(X²) = 6
• Var (X) = E (X2) - [E (X)]2 = 6 - 22 = 2
• Standard deviation of X =  √Var(X) =√2

• Therefore, E(X) = 2 and standard deviation = √2.