Problem 3 By next year, the stock you own ... [FREE SOLUTION]

Chapter 17: Problem 3

By next year, the stock you own has a $75 \%$ chance of being worth $\$ 1,000$ and a $25 \%$ probability of being worth $\$ 400 .$ What are the expected value and the variance? $\mathbf{A}$

Short Answer

Expert verified

Expected value is $850, and variance is 67500.

Step by step solution

Identify the Probabilities and Outcomes

The problem gives us two possible outcomes for the stock value with their respective probabilities: there is a 75% chance that the stock is worth $1,000, and a 25% chance that it is worth $400. We can express these probabilities and values as $ P(X = 1000) = 0.75 $ and $ P(X = 400) = 0.25 $.

Calculate the Expected Value

To find the expected value, we use the formula $ E(X) = \sum (x_i \times P(x_i)) $. This means we multiply each outcome by its probability and then sum these products: $ E(X) = 1000 \times 0.75 + 400 \times 0.25 $. Calculating this, we get: $ E(X) = 750 + 100 = 850 $. Thus, the expected value of the stock is $ \$850 $.

Calculate the Variance

Variance measures how much the outcomes spread out from the expected value. It is calculated using the formula $ Var(X) = \sum ((x_i - E(X))^2 \times P(x_i)) $. First, calculate the squared differences from the expected value: $(1000 - 850)^2 = 22500 $ and $(400 - 850)^2 = 202500 $. Next, calculate the weighted sum of these squared differences: $ Var(X) = 22500 \times 0.75 + 202500 \times 0.25 $. This simplifies to $ Var(X) = 16875 + 50625 = 67500 $. Therefore, the variance of the stock's value is 67500.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability

Probability is a measure of how likely an event is to occur. It is usually expressed as a number between 0 and 1, where 0 means the event is impossible and 1 indicates certainty. In this exercise, we can think of probability as a way to model uncertainty about future stock values.
In the given problem, the probability that the stock will be worth $1,000 is 0.75, or 75%. This outcome is more likely than the probability of the stock being worth $400, which is 0.25, or 25%.
Understanding the concept of probability can help us better prepare for various possible outcomes and make informed decisions.

Variance

Variance is a statistical measure that tells us how much a set of values, like possible stock prices, differs from the expected value. It helps us quantify the variability or the spread of the data points from the average outcome.
In this example, the variance is calculated using the formula:
\[Var(X) = \sum ((x_i - E(X))^2 \times P(x_i))\]We first find how far each potential stock price is from the expected value, square these differences, and multiply by their respective probabilities. This produces a value of 67,500. A higher variance indicates greater variability around the expected value, suggesting more uncertainty in future outcomes.

Probability Distributions

A probability distribution shows all the possible outcomes of a random variable, in this case, the future value of the stock, along with the probabilities of these outcomes. It provides a complete picture of all potential outcomes and their likelihoods.
For the stock example, we have a simple distribution with two outcomes:

The stock is worth $1,000 with a probability of 0.75.
The stock is worth $400 with a probability of 0.25.

This distribution helps us calculate both the expected value and variance by mapping out all possible scenarios that can occur. Understanding probability distributions is crucial for making predictions and managing risks.

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91影视

By next year, the stock you own has a \(75 \%\) chance of being worth \(\$ 1,000\) and a \(25 \%\) probability of being worth \(\$ 400 .\) What are the expected value and the variance? \(\mathbf{A}\)