Question

A survey was conducted among some first graders and one of the questions was how much money they spent on their school cafeteria. The random variable x represents the amount of money spent on a particular day with the corresponding probability P(X).

x 1 3 5 10 20
P(X) 0.16 ? 0.22 0.22 0.08


a) What is the probability that a 1st grader spends exactly 3 dollars?
b) Calculate the expected value, variance, and standard deviation of the random variable X?

Answer 1

Solution:

a).

$$P(X=3)= 1-(0.16+0.22+0.22+0.08)$$ $$=0.32$$

b).

$$E(X) = \sum xP(X=x)$$

=1(0.16)+3(0.32)+5(0.22)+10(0.22)+20(0.08)

$=6.02$

$$Var(x) = \sum x^2 P(x) - {(E(x))} ^2$$

$=1^2(0.16)+3^2(0.32)+5^2(0.22)$

$+10(0.22)+20(0.08)$

$$=62.54- {(6.02)}^2$$

= 26.2996

Standard deviation

$$Sd = \sqrt {var}$$

$$= \sqrt {26.2996} =5.1283$$