Question

1. random variables. given independent random variables with means and standard deviations as shown, find the mean and standard deviation of each of these variables:

a) ( x + 50 )
b) ( 10y )
c) ( x + 0.5y )
d) ( x - y )
e) ( x_1 + x_2 )
(chart: mean - x: 50, y: 100; sd - x: 8, y: 6)

Explanation:

Step1: Recall Mean and SD Properties

For a random variable \( X \) with mean \( \mu_X \) and SD \( \sigma_X \), and constant \( a, b \), and independent \( X, Y \):

• Mean of \( aX + bY \): \( \mu_{aX + bY} = a\mu_X + b\mu_Y \)
• Variance of \( aX + bY \): \( \sigma_{aX + bY}^2 = a^2\sigma_X^2 + b^2\sigma_Y^2 \) (since independent, covariance \( = 0 \))
• SD is square root of variance.

Part (a): \( X + 50 \)

Step2: Calculate Mean

\( \mu_{X + 50} = \mu_X + 50 \). Given \( \mu_X = 50 \), so \( \mu = 50 + 50 = 100 \).

Step3: Calculate SD

Adding a constant does not change variance/SD. \( \sigma_{X + 50} = \sigma_X = 8 \).

Part (b): \( 10Y \)

Step2: Calculate Mean

\( \mu_{10Y} = 10\mu_Y \). Given \( \mu_Y = 100 \), so \( \mu = 10 \times 100 = 1000 \).

Step3: Calculate SD

Variance of \( 10Y \): \( \sigma_{10Y}^2 = 10^2 \sigma_Y^2 = 100 \times 6^2 = 100 \times 36 = 3600 \).
SD: \( \sigma = \sqrt{3600} = 60 \).

Part (c): \( X + 0.5Y \)

Step2: Calculate Mean

\( \mu_{X + 0.5Y} = \mu_X + 0.5\mu_Y \).
Substitute \( \mu_X = 50 \), \( \mu_Y = 100 \):
\( \mu = 50 + 0.5 \times 100 = 50 + 50 = 100 \).

Step3: Calculate SD

Variance: \( \sigma_{X + 0.5Y}^2 = \sigma_X^2 + (0.5)^2\sigma_Y^2 \).
Substitute \( \sigma_X = 8 \) (so \( \sigma_X^2 = 64 \)), \( \sigma_Y = 6 \) (so \( \sigma_Y^2 = 36 \)):
\( \sigma^2 = 64 + 0.25 \times 36 = 64 + 9 = 73 \).
SD: \( \sigma = \sqrt{73} \approx 8.544 \).

Part (d): \( X - Y \)

Step2: Calculate Mean

\( \mu_{X - Y} = \mu_X - \mu_Y \).
Substitute \( \mu_X = 50 \), \( \mu_Y = 100 \):
\( \mu = 50 - 100 = -50 \).

Step3: Calculate SD

Variance: \( \sigma_{X - Y}^2 = \sigma_X^2 + (-1)^2\sigma_Y^2 = \sigma_X^2 + \sigma_Y^2 \) (since independent).
Substitute \( \sigma_X^2 = 64 \), \( \sigma_Y^2 = 36 \):
\( \sigma^2 = 64 + 36 = 100 \).
SD: \( \sigma = \sqrt{100} = 10 \).

Part (e): \( X_1 + X_2 \) (Assume \( X_1, X_2 \) are copies of \( X \), so \( \mu_{X_1} = \mu_{X_2} = 50 \), \( \sigma_{X_1} = \sigma_{X_2} = 8 \))

Step2: Calculate Mean

\( \mu_{X_1 + X_2} = \mu_{X_1} + \mu_{X_2} = 50 + 50 = 100 \).

Step3: Calculate SD

Variance: \( \sigma_{X_1 + X_2}^2 = \sigma_{X_1}^2 + \sigma_{X_2}^2 = 8^2 + 8^2 = 64 + 64 = 128 \).
SD: \( \sigma = \sqrt{128} = 8\sqrt{2} \approx 11.31 \).

Answer:

s (Summary):

Expression	Mean	Standard Deviation
\( 10Y \)	1000	60
\( X + 0.5Y \)	100	\( \sqrt{73} \approx 8.54 \)
\( X - Y \)	-50	10
\( X_1 + X_2 \)	100	\( 8\sqrt{2} \approx 11.31 \)

(For each part, substitute values into the mean/variance formulas as shown above.)