9.3. Practice Problems

1. Find the expectation of the number of heads in three tosses of a fair coin.

In this problem, we are asked to find the expectation for the number of heads when tossing a fair coin three times. Each coin toss can either result in heads (H) or tails (T). The outcomes can be described as follows: For three tosses, the possible outcomes are HHH, HHT, HTH, HTT, THH, THT, TTH, TTT. To find the expectation, we can use the concept of probability. The number of heads can take values ranging from 0 to 3. We can calculate the probability of each outcome and then use the formula for expectation, which is the sum of all possible values multiplied by their probabilities.

Imagine you are flipping a coin with your friends. If you flip it three times, what do you think the average number of heads will be? It's like expecting how many times you might get a heads when playing a game that involves flipping coins. If you keep track of the results over many games, the average number of heads gives you insight into the fairness of the coin.

2. Let 𝑋 be uniformly distributed on [2,4]. Find 𝐸(𝑋).

In this problem, we have a random variable 𝑋 that is uniformly distributed between the values of 2 and 4. For a uniform distribution, the expectation (mean) can be calculated by taking the average of the minimum and maximum values. The formula for expectation in a uniform distribution on the interval [a, b] is E(X) = (a + b) / 2. Hence, for this problem, plugging in our values gives E(X) = (2 + 4) / 2 = 3.

Think of a scenario where you are choosing a random number between 2 and 4, perhaps as if you are selecting a prize from a basket that has evenly distributed prizes labeled 2, 3, and 4. If you were to repeatedly pick numbers, you’d notice that over time, you’d average out to around 3, which represents the average prize.

3. Prove the linearity property of expectation for two arbitrary random variables.

This exercise involves proving that the expectation of a linear combination of two random variables is the same as the linear combination of their expectations. The linearity property states that E(aX + bY) = aE(X) + bE(Y) for any constants a and b. To prove this, we can start from the definitions of expectation and use the properties of summation and linearity in integrals or sums. This is fundamental in probability theory as it indicates that expectations can be calculated in a straightforward manner despite the potential complexity of the random variables themselves.

Consider you're an investor looking at two different stocks. You know the expected returns from each stock. If you were to invest different amounts in these stocks (say $1000 in Stock X and$1500 in Stock Y), the overall expected return isn't just a complex calculation but can simply be thought of as a weighted combination of the two expected returns based on your investments. The linearity helps you quickly estimate your potential returns.

4. If 𝑋 and 𝑌 are independent random variables with 𝐸(𝑋) = 3, 𝐸(𝑌) = 4, find 𝐸(2𝑆+3𝑌).

The problem provides us with the expectations of two independent random variables, X and Y, and asks us to find the expectation of a new random variable formed by a linear combination of these two random variables. Using the linearity of expectation, we can say that E(2X + 3Y) = 2E(X) + 3E(Y). Substituting the known expectations gives E(2X + 3Y) = 2(3) + 3(4) = 6 + 12 = 18.

Imagine you've started two side businesses – one selling lemonade and the other selling cookies. Based on your past sales data, you expect to make $3 per lemonade and$4 per cookie. If you plan to sell twice as many lemonades and three times as many cookies, your total expected revenue from both businesses would be easily computable using the expected values.

5. For a PDE problem 𝑢 = 𝐷𝑢 with uncertain initial condition, explain how 𝐸[𝑢(𝑥,𝑡)] can be used to derive a deterministic equation.

This problem connects the concept of expectation with partial differential equations (PDEs), particularly when dealing with uncertainty. In the equation provided, u corresponds to a solution of a PDE with an uncertain initial condition. By taking the expectation E[u(x, t)], we can average over the uncertainty, leading to a new equation that can often be interpreted deterministically. This simplifies analysis, allowing one to work with expected values instead of fully stochastic variables.

Consider predicting the temperature in a city where initial forecasts are uncertain due to weather patterns. If we average out those uncertainties (like sunny, rainy, cloudy), we can come up with a clearer, more reliable forecast that helps local authorities prepare for the expected weather conditions, ensuring better decision-making, such as issuing alerts or adjusting resource allocation.

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