4.8 Mean and Standard Deviation of Random Variables

• The expected value of a random variable is also known as its mean.


  True
• What is the formula for calculating the mean of a discrete random variable?

  \mu = \sum_{i = 1}^{n} x_{i} \cdot P(x_{i})</latex>
• The standard deviation of a random variable measures its variability
• What does the standard deviation of a random variable measure?

  The spread of values
• What is the formula for calculating the standard deviation of a discrete random variable?

  $\sigma =$$\sqrt{\sum_{i = 1}^{n} (x_{i} - \mu)^{2} \cdot P(x_{i})}$
• What does the mean of a discrete random variable represent?

  Average or central tendency
• Match the property with its interpretation:

  Mean ↔️ Represents the central tendency

  Standard Deviation ↔️ Represents the dispersion or spread
• What does the standard deviation of a random variable measure?

  Spread or variability
• What does the mean of a discrete random variable represent?

  Average or central tendency
• What is the probability density function (PDF) used for in continuous random variables?

  Calculating the mean
• Compare the properties of discrete and continuous random variables:

  1️⃣ Discrete values are countable

  2️⃣ Continuous values are uncountable

  3️⃣ Discrete uses PMF, continuous uses PDF

  4️⃣ Mean calculation is summation for discrete

  5️⃣ Mean calculation is integration for continuous
• A lower standard deviation indicates that the values of a discrete random variable are more concentrated around the mean.

  True
• Match the terms with their definitions:

  x_i ↔️ Possible values of the random variable

  P(x_i) ↔️ Probabilities of each value

  \mu ↔️ Mean of the random variable
• The standard deviation measures the variability
• What is the PDF of a uniform distribution on the interval [0, 2]?

  $f(x) =$$\frac{1}{2}$
• Match the properties with their definitions:

  Expected Value (Mean) ↔️ The average value of the random variable

  Standard Deviation ↔️ A measure of the spread of the random variable
• Discrete random variables have countable values.


  True
• Arrange the properties of a random variable in a table format:

  1️⃣ Property

  2️⃣ Expected Value (Mean)

  3️⃣ Standard Deviation

  4️⃣ Definition

  5️⃣ Formula

  6️⃣ Interpretation
• The mean of a random variable is the expected value of that variable.
• What does the standard deviation of a random variable measure?

  Spread around the mean
• The mean of a discrete random variable represents its central tendency.
• What is the formula for the mean of a continuous random variable?

  $\mu =$$\int_{ - \infty}^{\infty} x \cdot f(x) \, dx$
• The standard deviation of a discrete random variable is calculated as the square root of the sum of squared differences from the mean.
• The standard deviation of a discrete random variable measures the spread of values around its mean.


  True
• The sum of $(x_{i} - \mu)^{2} \cdot P(x_{i})$ for the discrete random variable $X$ is 0.61
• The formula for the standard deviation of a continuous random variable is $\sigma =$$\sqrt{\int_{ - \infty}^{\infty} (x - \mu)^{2} \cdot f(x) \, dx}$, where $f(x)$ is the PDF
• The standard deviation of a uniform distribution on $[0, 2]$ is approximately 0.408
• If $X$ has a mean of 5 and standard deviation of 2, what is the mean of $Y =$$3X - 1$?


  14
• The standard deviation of a random variable measures its spread
• The mean of a random variable represents its expected value.


  True
• The mean of a discrete random variable indicates the central tendency of its probability distribution.


  True
• The formula for the standard deviation of a discrete random variable involves the square root of a sum
• What does the standard deviation of a random variable measure?

  Spread or variability
• The mean of a discrete random variable is the sum of each possible value multiplied by its probability.


  True
• What does the mean of a random variable represent over many trials?

  Expected value
• The formula to calculate the standard deviation of a random variable is \sigma
• The formula to calculate the mean of a discrete random variable is \mu
• The formula to calculate the mean of a continuous random variable is \mu
• What does the standard deviation of a discrete random variable measure?

  Spread or variability
• The standard deviation of a discrete random variable measures its spread