Let Xn be the random variable that equals the number of...

Let $X_{n}$ be the random variable that equals the number of tails minus the number of heads when $n$ fair coins are flipped.
a) What is the expected value of $X_{n} ?$
b) What is the variance of $X_{n} ?$

Key Concepts

Random Variable

A random variable is a function that assigns a numerical value to each outcome in a sample space of a random experiment. It serves as a tool to quantify randomness and enables the application of probability theory to analyze outcomes numerically.

Expected Value

The expected value is a measure of the center or average of a random variable's probability distribution. It is essentially a weighted average of all possible values, with weights given by their corresponding probabilities, and provides insight into the long-run behavior of the random process.

Variance

Variance quantifies the spread or dispersion of a random variable's values around its expected value. By measuring the average squared deviation from the mean, variance provides an understanding of the level of uncertainty or variability inherent in the random process.

Linearity of Expectation

Linearity of expectation is a principle stating that the expected value of the sum of random variables is equal to the sum of their individual expected values, regardless of whether the variables are independent. This property is extremely useful in simplifying the calculation of expected values in complex scenarios.

Independent Trials

Independent trials refer to experiments where the outcome of one trial does not affect the outcome of another. This concept is crucial when analyzing processes that involve repeated random events, such as flipping coins, as it allows for straightforward computation of overall probabilities, expectations, and variances.

Transcript

00:01 All right, so here we're looking at letting x -n be equal to the number of tails, right? so let's say number of tails minus the number of heads when n fair coins are flipped.

00:23 Okay, so first we note that the probability of getting a heads is equal to the probability of getting a tails when we have a fair coin and that is equal to one half.

00:36 Okay, so then the expected value of, say, getting a tails, since this is bernoulli trial, is going to just be equal to the number of trials times the probability of getting a tails, right? and so we don't know what n is, but we know that the probability getting a tails is one -half, so this is equal to 1 over 2.

01:07 And this is going to be equal to the expected value for our heads as well.

01:17 Okay, so we can use this then to determine the expected value of x, right, which is this difference.

01:26 So since x is equal to tails minus heads, then the expected value of x is going to equal to the expected value of the tails minus the expected value of heads in these n number of trials.

01:45 Okay, so that is equal to n over 2 minus n over 2, which is zero.

01:57 Okay, and this was just part a.

02:03 So next we have a second part to this question, part b here.

02:07 And this is about the variance.

02:10 Okay, so first we'll note that variance for, bernouille trials is equal to the number of trials times the probability of success times the probability of failure, which is q or 1 minus p, depending on how you like to write it...

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