Respuesta :
Answer:
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
The expected value, the variance, and the standard deviation of the given random variable X are 1.8, 0.36, and 0.6 respectively and this can be determined by using the given data.
Given :
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.
The total number of outcomes are:
[tex]\rm \; ^5C_3=10[/tex]
The probability of random variable X are as follows:
X = 1 red marble
P(X = 1) = [tex]\rm \; ^3C_1 \times ^2C_2[/tex] = 0.3
X = 2 red marbles
P(X = 2) = [tex]\rm \; ^3C_2 \times ^2C_1[/tex] = 0.6
X = 3 red marbles
P(X = 3) = [tex]\rm \; ^3C_3[/tex] = 0.1
So, the expected value is given by:
[tex]\rm E(X) = \sum X_i \times P(X_i)[/tex]
[tex]= 1\times 0.3+2\times 0.6 + 3 \times 0.1[/tex]
= 1.8
Now, the variance is given by:
[tex]\rm Var(X) = \sum X^2_i \times P(X_i)-(E(X))^2[/tex]
[tex]=1^2\times 0.3+2^2\times 0.6+3^2 \times 0.1-1.8^2[/tex]
= 0.36
Now, the standard deviation is given by:
[tex]\rm \sigma=\sqrt{Var(X)}[/tex]
[tex]\rm \sigma = \sqrt{0.36}[/tex]
[tex]\sigma = 0.6[/tex]
For more information, refer to the link given below:
https://brainly.com/question/23091366
