Question

calculating the standard deviation of a discrete random variable
a popular streaming service surveyed all of the students at a school about the number of tv shows they streamed last friday night, then recorded the results.
let ( m ) = the number of tv shows streamed last friday night.

number of shows streamed	probability
1	0.203
2	0.326
3	0.187
4	0.174
5	0.015

calculate the standard deviation of ( m ).

Explanation:

Step1: Calculate the expected value \( E(M) \)

The formula for the expected value of a discrete random variable is \( E(M) = \sum_{i} m_i \cdot P(m_i) \), where \( m_i \) is the value of the random variable and \( P(m_i) \) is its corresponding probability.

\[

$$\begin{align*} E(M) &= 0 \cdot 0.095 + 1 \cdot 0.203 + 2 \cdot 0.326 + 3 \cdot 0.187 + 4 \cdot 0.174 + 5 \cdot 0.015 \\ &= 0 + 0.203 + 0.652 + 0.561 + 0.696 + 0.075 \\ &= 0.203 + 0.652 + 0.561 + 0.696 + 0.075 \\ &= 0.855 + 0.561 + 0.696 + 0.075 \\ &= 1.416 + 0.696 + 0.075 \\ &= 2.112 + 0.075 \\ &= 2.187 \end{align*}$$

\]

Step2: Calculate the variance \( \text{Var}(M) \)

The formula for the variance of a discrete random variable is \( \text{Var}(M) = \sum_{i} (m_i - E(M))^2 \cdot P(m_i) \).

First, calculate \( (m_i - E(M))^2 \) for each \( m_i \):

• For \( m = 0 \): \( (0 - 2.187)^2 = (-2.187)^2 = 4.782969 \), then \( 4.782969 \cdot 0.095 \approx 0.454382 \)
• For \( m = 1 \): \( (1 - 2.187)^2 = (-1.187)^2 = 1.408969 \), then \( 1.408969 \cdot 0.203 \approx 0.286021 \)
• For \( m = 2 \): \( (2 - 2.187)^2 = (-0.187)^2 = 0.034969 \), then \( 0.034969 \cdot 0.326 \approx 0.011400 \)
• For \( m = 3 \): \( (3 - 2.187)^2 = (0.813)^2 = 0.660969 \), then \( 0.660969 \cdot 0.187 \approx 0.123601 \)
• For \( m = 4 \): \( (4 - 2.187)^2 = (1.813)^2 = 3.286969 \), then \( 3.286969 \cdot 0.174 \approx 0.571933 \)
• For \( m = 5 \): \( (5 - 2.187)^2 = (2.813)^2 = 7.912969 \), then \( 7.912969 \cdot 0.015 \approx 0.118695 \)

Now, sum these values to get the variance:

\[

$$\begin{align*} \text{Var}(M) &\approx 0.454382 + 0.286021 + 0.011400 + 0.123601 + 0.571933 + 0.118695 \\ &= 0.454382 + 0.286021 + 0.011400 + 0.123601 + 0.571933 + 0.118695 \\ &= 0.740403 + 0.011400 + 0.123601 + 0.571933 + 0.118695 \\ &= 0.751803 + 0.123601 + 0.571933 + 0.118695 \\ &= 0.875404 + 0.571933 + 0.118695 \\ &= 1.447337 + 0.118695 \\ &= 1.566032 \end{align*}$$

\]

Step3: Calculate the standard deviation \( \sigma \)

The standard deviation is the square root of the variance, so \( \sigma = \sqrt{\text{Var}(M)} \).

\[
\sigma = \sqrt{1.566032} \approx 1.251
\]

Answer:

The standard deviation of \( M \) is approximately \( 1.25 \) (rounded to two decimal places) or more precisely \( \approx 1.251 \).