Question

a variable y of a finite population has the frequency distribution shown in the table. suppose a member is selected at random from the population and let y denote the value of the variable y for the member obtained. complete parts (a) through (d) below.
a. determine the probability distribution of the random variable y.
b. use random - variable notation to describe the events that y takes on the value 6, a value less than 6, and a value of at least 6.
the event that y takes on the value 6 can be represented as \\{y = 6\\}.
the event that y takes on a value less than 6 can be represented as \\{y < 6\\}.
the event that y takes on a value of at least 6 can be represented as \\{y\geq6\\}.
c. find p(y = 6), p(y < 6), and p(y\geq6).
p(y = 6)=.45 (type an integer or a decimal. do not round.)
p(y < 6)=\square (type an integer or a decimal. do not round.)

Explanation:

Step1: Calculate \(P(Y < 6)\)

The values of \(Y\) less than \(6\) are \(Y = 4\) and \(Y=5\).
Using the formula for the probability of a discrete random variable \(P(A)=\sum_{y\in A}P(Y = y)\), we have \(P(Y < 6)=P(Y = 4)+P(Y = 5)\).
Substituting the given probabilities \(P(Y = 4)=0.15\) and \(P(Y = 5)=0.15\), we get \(P(Y < 6)=0.15 + 0.15\).

Step2: Simplify the sum

\(0.15+0.15 = 0.3\)

Answer:

\(0.3\)