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 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\geq5\\}.
c. find p(y = 5), p(y<5), and p(y\geq5).
p(y = 5)=.6 (type an integer or a decimal. do not round.)
p(y<5)=\square (type an integer or a decimal. do not round.)

Explanation:

Step1: Calculate \(P(Y < 5)\)

The values of \(Y\) less than \(5\) are \(Y = 3\) and \(Y=4\).

$$P(Y < 5)=P(Y = 3)+P(Y = 4)$$

$$P(Y < 5)=0.1 + 0.1$$

$$P(Y < 5)=0.2$$

Step2: Calculate \(P(Y\geq5)\)

We know that the sum of all probabilities in a probability distribution is \(1\).

$$P(Y\geq5)=1 - P(Y < 5)$$

$$P(Y\geq5)=1-0.2$$

$$P(Y\geq5) = 0.8$$

Answer:

a. The probability distribution is as given in the table (already provided in the problem statement).
b. The event \(Y = 5\) is \(\{Y = 5\}\), \(Y<5\) is \(\{Y = 3,Y = 4\}\), \(Y\geq5\) is \(\{Y = 5,Y = 6\}\)
c. \(P(Y = 5)=0.6\), \(P(Y < 5)=0.2\), \(P(Y\geq5)=0.8\)