Question

Alex tracked how many cups of coffee he drank each day for a month and then compiled the results into a probability distribution, as shown below.

Number of Cups of Coffee, upper X 0 1 2 3 4
Relative Frequency, upper P left parenthesis upper X right parenthesis 0.024 0.423 0.291 0.172 0.090
Question
Calculate the expected value of the number of cups of coffee Alex drinks each day.

(1 point)
Responses

0.200 cups
0.200 cups

0.163 cups
0.163 cups

0.976 cups
0.976 cups

1.881 cups

Answer 1

To calculate the expected value (E[X]) of the number of cups of coffee Alex drinks each day, we can use the formula for expected value, which is:

\[
E[X] = \sum (x_i \cdot P(X = x_i))
\]

where \(x_i\) is the number of cups of coffee and \(P(X = x_i)\) is the corresponding relative frequency.

Using the data provided:

- For 0 cups: \(0 \cdot 0.024 = 0\)
- For 1 cup: \(1 \cdot 0.423 = 0.423\)
- For 2 cups: \(2 \cdot 0.291 = 0.582\)
- For 3 cups: \(3 \cdot 0.172 = 0.516\)
- For 4 cups: \(4 \cdot 0.090 = 0.360\)

Now, we sum these values to calculate the expected value:

\[
E[X] = 0 + 0.423 + 0.582 + 0.516 + 0.360
\]

Calculating the sum step-by-step:

1. \(0 + 0.423 = 0.423\)
2. \(0.423 + 0.582 = 1.005\)
3. \(1.005 + 0.516 = 1.521\)
4. \(1.521 + 0.360 = 1.881\)

Thus, the expected value of the number of cups of coffee Alex drinks each day is:

\[
E[X] = 1.881 \text{ cups}
\]

The correct answer is:

**1.881 cups**