Problem 62 - Applied Statistics and Probability for Engineers

Question

\(X\) and \(Y\) are independent, normal random variables with \(E(X)=0, V(X)=4, E(Y)=10,\) and \(V(Y)=9\) Determine the following: (a) \(E(2 X+3 Y)\) (b) \(V(2 X+3 Y)\) (c) \(P(2 X+3 Y<30)\) (d) \(P(2 X+3 Y<40)\)

Step-by-Step Solution

Answer

(a) 30 (b) 97 (c) 0.5 (d) 0.845

Step 1: Calculate E(2X + 3Y)

To find the expected value of a linear combination of independent random variables, use the formula: \[ E(aX + bY) = aE(X) + bE(Y) \]Here, we have:\[ E(2X + 3Y) = 2 \cdot E(X) + 3 \cdot E(Y) \]Since \( E(X) = 0 \) and \( E(Y) = 10 \), we find:\[ E(2X + 3Y) = 2 \cdot 0 + 3 \cdot 10 = 30 \].

Step 2: Calculate V(2X + 3Y)

For independent variables \( X \) and \( Y \), the variance of a linear combination is given by:\[ V(aX + bY) = a^2V(X) + b^2V(Y) \]Using \( V(X) = 4 \) and \( V(Y) = 9 \), we have:\[ V(2X + 3Y) = 2^2 \, V(X) + 3^2 \, V(Y) = 4 \cdot 4 + 9 \cdot 9 = 16 + 81 = 97 \].

Step 3: Standardize the distribution of 2X + 3Y

Since \( 2X + 3Y \) is a normal random variable with \( E(2X + 3Y) = 30 \) and \( V(2X + 3Y) = 97 \), its standard deviation is \( \sqrt{97} \).To compute probabilities, convert \( 2X + 3Y \) to a standard normal variable \( Z \) using:\[ Z = \frac{(2X + 3Y) - 30}{\sqrt{97}} \].

Step 4: Calculate P(2X + 3Y < 30)

For this probability, we seek:\[ P(2X + 3Y < 30) = P\left(Z < \frac{30 - 30}{\sqrt{97}}\right) = P(Z < 0) \]For a standard normal distribution, \( P(Z < 0) = 0.5 \).

Step 5: Calculate P(2X + 3Y < 40)

Now compute:\[ P(2X + 3Y < 40) = P\left(Z < \frac{40 - 30}{\sqrt{97}}\right) \]Calculate the Z-score:\[ Z = \frac{10}{\sqrt{97}} \approx 1.015259 \]Using the standard normal distribution table, find \( P(Z < 1.015259) \approx 0.845 \).

Key Concepts

Normal distribution

The concept of a normal distribution is fundamental in probability theory and statistics. It describes how the values of a variable are distributed. A normal distribution, also frequently referred to as a Gaussian distribution, is symmetric, centered around its mean, and characterized by its bell-shaped curve. It is defined by two parameters: the mean (\( \mu \)) and the variance (\( \sigma^2 \)). The mean represents the average or expected value, while the variance indicates the distribution's spread or how far values typically differ from the mean.

• The curve is symmetric about the mean.
• The mean, median, and mode of a normal distribution are all equal.
• Approximately 68% of the data falls within one standard deviation (\( \sigma \)) from the mean, 95% within two, and 99.7% within three, forming the empirical rule or 68-95-99.7 rule.

The normal distribution is significant due to the Central Limit Theorem, which states that the distribution of the sum of many independent, identically distributed variables approaches a normal distribution, regardless of the original distribution of the variables.

Expected value

The expected value, often called the mean, is a measure of the center of a probability distribution. It gives us the long-term average if an experiment is repeated many times. For a normal random variable, the expected value is simply the arithmetic mean of all possible values.
For a random variable \( X \), the expected value is denoted as \( E(X) \). When dealing with linear combinations of independent random variables, such as \( 2X + 3Y \), you can calculate the expected value by using a linear expectation rule:

• \( E(aX + bY) = a \cdot E(X) + b \cdot E(Y) \)
• This property shows that expectation is a linear operator.
• It is important to ensure the random variables involved are independent, as this assumption simplifies calculations significantly.

In our example, \( E(2X + 3Y) \) was calculated by multiplying the coefficients by their respective expected values, resulting in \( 30 \). This expected value tells us that over many trials, the average of \( 2X + 3Y \) will converge to \( 30 \).

Variance

Variance is another crucial concept in probability theory, representing the spread or dispersion of a set of values. It measures the average squared deviation of each value from the mean, indicating how much the values in a distribution typically vary. For a random variable \( X \), the variance is denoted as \( V(X) \) or sometimes \( \sigma^2 \).
Calculating variance for a linear combination of independent variables, like \( 2X + 3Y \), involves squared coefficients:

• \( V(aX + bY) = a^2 \cdot V(X) + b^2 \cdot V(Y) \)
• This property implies that variance is not a linear operator, due to the squaring of coefficients.
• It’s crucial that the random variables are independent to directly apply this formula without any additional correlation adjustments.

In the given problem, the variance for \( 2X + 3Y \) was calculated as \( 97 \), showing a substantial spread. This variance provides an understanding of how much the values are expected to deviate from their mean.

Standard normal distribution

The standard normal distribution is a specific kind of normal distribution where the mean (\( \mu \)) is \( 0 \) and the variance (\( \sigma^2 \)) is \( 1 \). It is characterized by a bell-shaped curve and is denoted by \( Z \), which is the standard score. This distribution is widely used due to its simplicity in statistical calculations.
Transforming a normal distribution to a standard normal distribution involves computing the Z-score, which standardizes any normal random variable to have these standard properties:

• The formula for the Z-score is:\( Z = \frac{(X - \mu)}{\sigma} \)
• The transformation helps compare different distributions and calculate probabilities using standard normal distribution tables.
• The probabilities associated with Z-scores provide the likelihood of a random variable being less than or more than a certain value.

For instance, the problem requires converting the distribution of \( 2X + 3Y \) into a standard normal variable to calculate probabilities like \( P(2X + 3Y < 30) \). The transformation allows us to use standardized tables, rendering calculations straightforward.