Question

Alex tracked the number of cups of coffee he drank each day for the following month and created a new probability distribution.

Number of Cups of Coffee, upper X 0 1 2 3 4
Relative Frequency, upper P left parenthesis upper X right parenthesis 0.035 0.579 0.286 0.080 0.020
According this new data, Alex is expected to drink 1.471 cups per day.

Question
Calculate upper S left parenthesis upper X right parenthesis of Alex’s new probability distribution.

(1 point)
Responses

0.599
0.599

0.774
0.774

1.471
1.471

4.223

Answer 1

To calculate the variance \( S(X) \) of the probability distribution, we need to follow these steps:

1. **Calculate the expected value \( E(X) \)**. It is already given that \( E(X) = 1.471 \).

2. **Calculate the expected value of \( X^2 \)**:
\[
E(X^2) = \sum (x^2 \cdot P(X = x))
\]
We calculate \( x^2 \) and \( P(X) \) for each value:
- For \( x = 0 \): \( 0^2 \cdot 0.035 = 0 \)
- For \( x = 1 \): \( 1^2 \cdot 0.579 = 0.579 \)
- For \( x = 2 \): \( 2^2 \cdot 0.286 = 4 \cdot 0.286 = 1.144 \)
- For \( x = 3 \): \( 3^2 \cdot 0.080 = 9 \cdot 0.080 = 0.720 \)
- For \( x = 4 \): \( 4^2 \cdot 0.020 = 16 \cdot 0.020 = 0.320 \)

Adding these up, we have:
\[
E(X^2) = 0 + 0.579 + 1.144 + 0.720 + 0.320 = 2.763
\]

3. **Calculate the variance \( Var(X) \)** using the formula:
\[
Var(X) = E(X^2) - (E(X))^2
\]
Substituting the values we found:
\[
Var(X) = 2.763 - (1.471)^2
\]
Calculate \( (1.471)^2 \):
\[
(1.471)^2 = 2.167041
\]

Thus,
\[
Var(X) = 2.763 - 2.167041 = 0.595959 \approx 0.596
\]

4. **Standard deviation \( S(X) \)** is the square root of the variance:
\[
S(X) = \sqrt{Var(X)} = \sqrt{0.596} \approx 0.774
\]

Therefore, the correct answer is:
\[
\boxed{0.774}
\]