Question

If a random sample of 300 children is selected, let X be the number of these children who have been diagnosed with ASD. What distribution does X follow? What is the expected value and standard deviation of X?

Answer 1

The random variable X follows a Binomial distribution.

`E(X)=X\\SD(X)=\sqrt{(X(300-X))/(300)}`

Explanation:

The random variable X defined as the number of children who have been diagnosed with ASD.

The random sample of children selected is of size n = 300.

The probability of children diagnosed with ASD is,
`P(X)=p=(X)/(300)`.

A children diagnosed with ASD is independent of all the others.

The random variable X follows a Binomial distribution.

`X\sim Bin(n=300, p=(X)/(300))`

The expected value of X is:

`E(X)=np=300* (X)/(300)=X`

The standard deviation of X is:

`SD(X)=√(np(1-p))=\sqrt{300* (X)/(300)[1-(X)/(300)]}=\sqrt{(X(300-X))/(300)}`