Answer to Question #107054 in Statistics and Probability for Ntando

Question

Consider the information below to answer questions 1 and 2.
A Speech Therapist knows that the number of children that consults with her on any given day is
given by the following discrete probability distribution. Let X be the number of children consulting
with the Speech Therapist on any given day and P.X D x/ be the associated probability.

Answer 1

Consider the information below to answer questions 1 and 2. A Speech Therapist knows that the number of children that consults with her on any given day is given by the following discrete probability distribution. Let X be the number of children consulting with the Speech Therapist on any given day and P(X = x) be the associated probability.

$$\begin{matrix} x & 1 & 2 & 3 & 4 & 5 \\ P(X=x) & 0.1 & 0.15 & 0.2 & 0.3 & 0.25 \end{matrix}$$

Question 1

Which of the following statements is incorrect?

(1) P that the Therapist will consult with at least one child is 1.

(2) P that the Therapist will consult with 1 child on any given day is 0.1.

(3) P that the Therapist will consult with no children on any given day is 0.

(4) P that the Therapist will consult with more than 5 children on any given day is 0.

(5) The expected number of children consulting with the Therapist on any given day is 4.

Solution

$$E(X)=0.1(1)+0.15(2)+0.2(3)+0.3(4)+0.25(5)=$$ $$3.45\not=4$$

(5) The expected number of children consulting with the Therapist on any given day is 4.

Question 2

What is the standard deviation of X?

(1) 11.01

(2) 3.45

(3) 3.32

(4) 1.65

(5) 1.28

Solution

$$E(X^2)=0.1(1)^2 +0.15(2)^2 +0.2(3)^2+0.3(4)^2+0.25(5)^2=$$ $$=13.55$$

$$Var(X)=\sigma^2=E(X^2)-(E(X))^2=$$ $$13.55-(3.45)^2=1.6475$$

$$\sigma=\sqrt{1.6475}\approx1.28$$

(5) 1.28