Find The Expected Value Of The Above Random Variable

The expected value of a random variable is a fundamental concept in probability theory, statistics, and decision theory. It represents the average value we would expect to obtain if we were to repeat an experiment associated with the random variable a large number of times. Finding the expected value is crucial for making informed decisions in various fields, from finance and gambling to engineering and scientific research. It helps us quantify the central tendency of a random variable and understand the long-term behavior of probabilistic events.

Understanding Random Variables

Before delving into calculating the expected value, it's important to define what a random variable is. A random variable is a variable whose value is a numerical outcome of a random phenomenon. Random variables can be discrete or continuous.

• Discrete Random Variable: A discrete random variable is one whose value can only take on a finite number of values or a countably infinite number of values. These values are often integers, representing counts or categories. Examples include the number of heads when flipping a coin four times (0, 1, 2, 3, or 4), the number of defective items in a batch of 100, or the number of customers arriving at a store in an hour.

• Continuous Random Variable: A continuous random variable is one whose value can take on any value within a given range. These values are real numbers and can include fractions and decimals. Examples include height, weight, temperature, or the time it takes to complete a task.

Understanding the type of random variable is essential because the method for calculating the expected value differs slightly for discrete and continuous random variables.

Expected Value of a Discrete Random Variable

The expected value of a discrete random variable, denoted as E(X) or μ (mu), is calculated by summing the product of each possible value of the random variable and its corresponding probability. The formula is:

E(X) = Σ [x * P(x)]

Where:

• X is the random variable.
• x represents the possible values that the random variable can take.
• P(x) is the probability of the random variable taking the value x.
• Σ represents the summation over all possible values of x.

Steps to Calculate the Expected Value of a Discrete Random Variable:

1. Identify all possible values (x) of the random variable. This involves understanding the scenario and determining the range of outcomes.
2. Determine the probability (P(x)) associated with each value. This might involve using probability distributions, counting techniques, or given data.
3. Multiply each value (x) by its corresponding probability (P(x)). This calculates the weighted average of each outcome.
4. Sum all the products obtained in step 3. This gives the expected value of the random variable.

Example 1: Flipping a Coin

Let's consider a simple example: flipping a fair coin twice. The random variable X is the number of heads obtained.

• Possible values of X: 0, 1, 2
• Probabilities:
  • P(X = 0) = 1/4 (Tail, Tail)
  • P(X = 1) = 2/4 = 1/2 (Head, Tail or Tail, Head)
  • P(X = 2) = 1/4 (Head, Head)

Now, we can calculate the expected value:

E(X) = (0 * 1/4) + (1 * 1/2) + (2 * 1/4) = 0 + 1/2 + 1/2 = 1

Therefore, the expected number of heads when flipping a fair coin twice is 1.

Example 2: Rolling a Die

Let's consider rolling a fair six-sided die. The random variable X is the number that appears on the die.

• Possible values of X: 1, 2, 3, 4, 5, 6
• Probabilities: Since the die is fair, each outcome has an equal probability of 1/6.
  • P(X = 1) = 1/6
  • P(X = 2) = 1/6
  • P(X = 3) = 1/6
  • P(X = 4) = 1/6
  • P(X = 5) = 1/6
  • P(X = 6) = 1/6

Now, we can calculate the expected value:

E(X) = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6) = (1 + 2 + 3 + 4 + 5 + 6) / 6 = 21 / 6 = 3.5

Therefore, the expected value when rolling a fair six-sided die is 3.5. This means that if you roll the die many times, the average of the numbers you roll will approach 3.5.

Example 3: Lottery Ticket

Suppose you buy a lottery ticket for $1. The probability of winning $1000 is 1/10000, and the probability of winning nothing is 9999/10000. What is the expected value of your lottery ticket?

• Possible values of X: $1000 - $1 = $999 (win), -$1 (lose)
• Probabilities:
  • P(X = $999) = 1/10000
  • P(X = -$1) = 9999/10000

Now, we can calculate the expected value:

E(X) = ($999 * 1/10000) + (-$1 * 9999/10000) = $999/10000 - $9999/10000 = -$9000/10000 = -$0.90

Therefore, the expected value of the lottery ticket is -$0.90. This means that on average, you are expected to lose $0.90 for every lottery ticket you buy. This is why lotteries are generally not a good investment.

Expected Value of a Continuous Random Variable

The expected value of a continuous random variable is calculated using integration. The formula is:

E(X) = ∫ [x * f(x)] dx

Where:

• X is the random variable.
• x represents the possible values that the random variable can take.
• f(x) is the probability density function (PDF) of the random variable.
• ∫ represents the integration over the entire range of possible values of x.

Steps to Calculate the Expected Value of a Continuous Random Variable:

1. Identify the probability density function (PDF) f(x) of the random variable. This function describes the relative likelihood of the random variable taking on a given value.
2. Determine the range of possible values for the random variable. This is the interval over which the integration will be performed.
3. Multiply the random variable (x) by its PDF (f(x)). This calculates the weighted value of each possible outcome.
4. Integrate the product obtained in step 3 over the range of possible values. This gives the expected value of the random variable.

Example 1: Uniform Distribution

Suppose X is a continuous random variable that is uniformly distributed between 0 and 1. The PDF is f(x) = 1 for 0 ≤ x ≤ 1, and f(x) = 0 otherwise.

To find the expected value:

E(X) = ∫ [x * f(x)] dx from 0 to 1 E(X) = ∫ [x * 1] dx from 0 to 1 E(X) = ∫ x dx from 0 to 1 E(X) = [x^2 / 2] from 0 to 1 E(X) = (1^2 / 2) - (0^2 / 2) = 1/2

Therefore, the expected value of a continuous random variable uniformly distributed between 0 and 1 is 0.5.

Example 2: Exponential Distribution

Suppose X is a continuous random variable that follows an exponential distribution with parameter λ = 2. The PDF is f(x) = 2e^(-2x) for x ≥ 0, and f(x) = 0 otherwise.

To find the expected value:

E(X) = ∫ [x * f(x)] dx from 0 to ∞ E(X) = ∫ [x * 2e^(-2x)] dx from 0 to ∞

This integral requires integration by parts. Let u = x and dv = 2e^(-2x) dx. Then du = dx and v = -e^(-2x).

E(X) = [-xe^(-2x)] from 0 to ∞ + ∫ [e^(-2x)] dx from 0 to ∞ E(X) = 0 + [-1/2 * e^(-2x)] from 0 to ∞ E(X) = 0 - (-1/2) = 1/2

Therefore, the expected value of a continuous random variable following an exponential distribution with parameter λ = 2 is 0.5.

Properties of Expected Value

The expected value has several important properties that make it a powerful tool in probability and statistics:

1. Linearity: For any constants a and b, and random variables X and Y:

   • E(aX + b) = aE(X) + b
   • E(X + Y) = E(X) + E(Y)

2. Expected Value of a Constant: The expected value of a constant is the constant itself:

   • E(c) = c, where c is a constant.

3. Independence: If X and Y are independent random variables, then:

   • E(XY) = E(X)E(Y)

These properties can simplify the calculation of expected values in complex scenarios. For instance, if you need to find the expected value of a sum of independent random variables, you can simply sum their individual expected values.

Applications of Expected Value

The concept of expected value has wide-ranging applications in various fields:

• Finance: In finance, expected value is used to evaluate the potential profitability of investments. Investors use expected return to compare different investment opportunities and make informed decisions about where to allocate their capital. For example, the expected value can be used to analyze the potential returns from stocks, bonds, or real estate investments, taking into account the probabilities of different market scenarios.

• Gambling: Expected value is a crucial concept in gambling. It helps players understand the odds of winning and the potential payouts. A game with a negative expected value means that, on average, the player will lose money in the long run. Casinos rely on the fact that most games have a negative expected value for the player.

• Insurance: Insurance companies use expected value to calculate premiums. They estimate the probability of different events occurring (e.g., car accidents, house fires) and the associated costs. The premium is set to cover the expected cost of these events, plus a profit margin for the insurance company.

• Decision Theory: Expected value is a key component of decision theory, which provides a framework for making optimal decisions under uncertainty. By calculating the expected value of different actions, decision-makers can choose the option that maximizes their expected payoff.

• Engineering: In engineering, expected value is used to analyze the reliability and performance of systems. For example, engineers might use expected value to estimate the average lifespan of a component or the expected cost of maintenance.

• Scientific Research: Researchers use expected value to analyze data and draw conclusions from experiments. It helps them quantify the central tendency of a variable and understand the variability around that central tendency.

Common Mistakes to Avoid

When calculating the expected value, it's important to avoid some common mistakes:

• Incorrectly Identifying Possible Values: Ensure that you have identified all possible values of the random variable. Missing a value can lead to an inaccurate expected value.
• Incorrectly Calculating Probabilities: The probabilities associated with each value must be accurate. If the probabilities are incorrect, the expected value will also be incorrect. Remember that the sum of all probabilities must equal 1.
• Mixing Up Discrete and Continuous Formulas: Using the discrete formula for a continuous random variable (or vice versa) will result in an incorrect answer. Make sure you are using the appropriate formula for the type of random variable you are dealing with.
• Forgetting to Weight Values by Probabilities: The expected value is a weighted average, so it's essential to multiply each value by its corresponding probability. Forgetting to do this will give you a simple average, which is not the same as the expected value.
• Ignoring Negative Values: If the random variable can take on negative values (e.g., losses), be sure to include those negative values in the calculation. Ignoring negative values will skew the expected value.

Advanced Concepts Related to Expected Value

While the basic concept of expected value is straightforward, there are several advanced concepts that build upon it:

• Variance and Standard Deviation: Variance and standard deviation measure the spread or dispersion of a random variable around its expected value. Variance is the expected value of the squared difference between the random variable and its expected value, while standard deviation is the square root of the variance.

• Conditional Expectation: Conditional expectation is the expected value of a random variable given that another event has occurred. It is a powerful tool for analyzing relationships between random variables.

• Law of Large Numbers: The law of large numbers states that as the number of trials of a random experiment increases, the average of the results will converge to the expected value. This is why the expected value is often interpreted as the long-run average.

• Central Limit Theorem: The central limit theorem states that the distribution of the sum (or average) of a large number of independent and identically distributed random variables will be approximately normal, regardless of the underlying distribution of the individual random variables. This theorem is fundamental to statistical inference.

Conclusion

Finding the expected value of a random variable is a fundamental skill in probability, statistics, and decision-making. Whether dealing with discrete or continuous random variables, understanding the underlying concepts and applying the appropriate formulas is crucial. By carefully identifying possible values, calculating probabilities, and weighting values accordingly, one can accurately determine the expected value and make informed decisions based on probabilistic outcomes. Avoiding common mistakes and exploring advanced concepts can further enhance one's ability to utilize expected value in complex scenarios. The ability to calculate and interpret expected value is a valuable asset in a wide range of fields, from finance and gambling to engineering and scientific research.