Question

a. determine the probability distribution of the random variable y.

(tvpe integers or decimals. do not round.)

b. use random - variable notation to describe the events that y takes on the value 5, a value less than 5, and a value of at least 5.

the event that y takes on the value 5 can be represented as \\( \\{ y = 5 \\} \\).

the event that y takes on a value less than 5 can be represented as \\( \\{ y < 5 \\} \\).

the event that y takes on a value of at least 5 can be represented as \\( \\{ y \geq 5 \\} \\).

c. find \\( p ( y = 5 ) \\), \\( p ( y < 5 ) \\), and \\( p ( y \geq 5 ) \\).

\\( p ( y = 5 ) = \\).6 (type an integer or a decimal. do not round.)

\\( p ( y < 5 ) = \\).2 (type an integer or a decimal. do not round.)

\\( p ( y \geq 5 ) = \\).8 (type an integer or a decimal. do not round.)

d. construct a probability histogram for the random variable y. choose the correct histogram below.

Explanation:

Step1: Analyze the probability distribution table

We are given the probability distribution of the random variable \(Y\) as \(P(Y = 3)=0.1\), \(P(Y = 4)=0.1\), \(P(Y = 5)=0.6\), \(P(Y = 6)=0.2\)

Step2: Calculate \(P(Y = 5)\)

From the table, directly read the value of \(P(Y = 5)\). So \(P(Y = 5)=0.6\)

Step3: Calculate \(P(Y<5)\)

\(P(Y<5)\) means \(P(Y = 3)+P(Y = 4)\). Using the formula \(P(A\cup B)=P(A)+P(B)\) (since \(Y = 3\) and \(Y = 4\) are mutually - exclusive events).
\(P(Y<5)=P(Y = 3)+P(Y = 4)=0.1 + 0.1=0.2\)

Step4: Calculate \(P(Y\geq5)\)

\(P(Y\geq5)\) means \(P(Y = 5)+P(Y = 6)\). Using the formula \(P(A\cup B)=P(A)+P(B)\) (since \(Y = 5\) and \(Y = 6\) are mutually - exclusive events).
\(P(Y\geq5)=P(Y = 5)+P(Y = 6)=0.6+0.2 = 0.8\)

Answer:

\(P(Y = 5)=0.6\), \(P(Y<5)=0.2\), \(P(Y\geq5)=0.8\)