Let N be a random variable defined as the number of hours slept per night by college seniors. Let’s suppose no one sleeps less than four hours or more than eight hours per night. The following probability function is given for N.

Question

f(N) = {0.15 for n = 4, 0.23 for n = 5, 0.19 for n = 6, 0.24 for n = 7, and 0.19 for n = 8}
What is the expected value for N? Give your answer as a decimal.

Answer 1

To calculate the expected value \( E(N) \) of the random variable \( N \), we use the formula for expected value of a discrete random variable:

\[ E(N) = \sum_{n} n \cdot f(n) \]

We have the probability function given as follows:

• \( f(4) = 0.15 \)
• \( f(5) = 0.23 \)
• \( f(6) = 0.19 \)
• \( f(7) = 0.24 \)
• \( f(8) = 0.19 \)

Now we can compute \( E(N) \):

\[ E(N) = 4 \cdot 0.15 + 5 \cdot 0.23 + 6 \cdot 0.19 + 7 \cdot 0.24 + 8 \cdot 0.19 \]

Calculating each term:

1. \( 4 \cdot 0.15 = 0.60 \)
2. \( 5 \cdot 0.23 = 1.15 \)
3. \( 6 \cdot 0.19 = 1.14 \)
4. \( 7 \cdot 0.24 = 1.68 \)
5. \( 8 \cdot 0.19 = 1.52 \)

Now, sum these values:

\[ E(N) = 0.60 + 1.15 + 1.14 + 1.68 + 1.52 \]

Calculating the sum step by step:

• \( 0.60 + 1.15 = 1.75 \)
• \( 1.75 + 1.14 = 2.89 \)
• \( 2.89 + 1.68 = 4.57 \)
• \( 4.57 + 1.52 = 6.09 \)

Thus, the expected value of \( N \) is:

\[ E(N) = 6.09 \]

So, the expected value for \( N \) is \( \boxed{6.09} \).