Answer to Question #264115 in Statistics and Probability for Nathan

Question

An experiment consists of rolling two fair six-sided dice, one which is red and the other blue. A random variable X is defined such that it equals 0 if the two dice show the same number, 1 if the number on the blue die is bigger than the number on the red die and 2 if the number on the red die is bigger than the number on the blue die. Consider the following statements and choose the correct alternative from a) to e) below.

I. P( X < 2 ) = 0.5833

II. E(X) = 1.25

III. The median is 1

a.) I, II and III are all true

b.) Only I and III are true

c.) Only II and III are true

d.) Only I and II are true

e.) Only III is true

Answer 1

The discrete random variable $X$ can take on the values $0,1,2$. The probabilities that the random variable X take on these values are given below,

p(X=0)=6/36=1/6since only 6 out of 36 values have the same number on the blue and red die.

p(X=1)=15/36=5/12 since 15 out of 36 values have a number on the blue die being bigger than the number on the red die.

p(x=2)=15/36=5/12 since 15 out of 36 values have a number on the red die being bigger than the number on the blue die.

Now, we need to verify the given statements

i)

$p(x\lt 2)=p(x=0)+p(x=1)=1/6+5/12=7/12= 0.5833333$

ii)

$E(X)=\sum^2_{x=0}x*p(X=x)=(0*1/6)+(1*5/12)+(2*5/12)=5/12+10/12=15/12=1.25$

iii)

To check whether for the median we enter the following commands in R

x=c(0,1,2)

x

median(x)

The output for these set of commands is

> median(x)

[1] 1

Therefore, the median is 1

From the alternatives given we can see that option a) is the correct one since the properties stated are true.