Calculate the expected value, the variance, and the standard deviation of the given random variable X. (Round all answers to two decimal places.) X is the number of red marbles that Suzan has in her hand after she selects three marbles from a bag containing three red marbles and two green ones.

E (X) = 1.8

Var (X) = 0.36

σ = 0.6

Step-by-step explanation:

Solution:-

- Denote the random variable X : is the number of red marbles that Suzan has in her hand after she selects three marbles.

- Total sample space (bag) have the following quantity of colored marbles:

Bag : { 3 Red, 2 Green }

- Suzan selects three marbles from the bag. The Event (X) defines the number of red marbles out of 3.

- The total number of outcomes / selections for randomly selecting 3 balls from the bag:

All outcomes = 5 C 3 = 10

- The probability distribution of the random variable X, we will use combinations to determine the required probabilities:

X = 1 red marble:

P (X = 1) : Suzan chooses 1 Red marble from the available 3 red marble and 2 green marbles.

P (X = 1) = [ 3C1*2C2 ] / all outcomes = (3*1) / 10 = 0.3

X = 2 red marble:

P (X = 2) : Suzan chooses 2 Red marble from the available 3 red marble and 1 green marbles.

P (X = 2) = [ 3C2*2C1 ] / all outcomes = (3*2) / 10 = 0.6

X = 3 red marble:

P (X = 3) : Suzan chooses 3 Red marble from the available 3 red marble.

P (X = 3) = [ 3C3] / all outcomes = (1) / 10 = 0.1

- The probability distribution is as follows:

X : 1 2 3

P (X) : 0.3 0.6 0.1

- The expected value E (X) for the given random variable X is:

E (X) = ∑Xi*P (Xi)

= 1*0.3 + 2*0.6 + 3*0.1

=1.8

- The variance Var (X) for the given random variable X is:

Var (X) = ∑Xi^2*P (Xi) - [ E (X) ] ^2

= 1^2*0.3 + 2^2*0.6 + 3^2*0.1 - 1.8^2

= 0.36

- The standard deviation for the given random variable X is:

σ = √Var (X)

σ = √0.36

σ = 0.6