A random variable quantifies chance events, and its **probability distribution** assigns a likelihood to each of its values.

The distribution contains all the information we have about the random variable, so it is a very important concept in probability theory.

In this quiz, we'll start with distributions of discrete random variables and then move on to the continuous case.

Roll a die in the shape of an icosahedron, a solid figure having twenty faces that are equilateral triangles.

Let \( X \) be the random variable that assigns a die roll its value; the range of \(X\) is equal to the set \( \{ 1, 2, \dots, 19, 20 \}.\) If the die is fair, what is \( P(X = n) ? \)

The **probability distribution** of the icosahedron die is called **uniform** because all of the likelihoods for the possible die rolls are the same.

There are plenty of nonuniform examples. Consider flipping a fair coin ten times in a row. Let \( N\) be the number of heads in such a flip sequence.

Find a formula for \(P(N = n) \) for \( n = 0, 1, 2 , \dots, 10, \) the random variable's possible values.

**Hint:** The number of ways to choose \( k \) objects from a set of \( n \) is \[{n \choose k} = \frac{n!}{k!(n-k)!},\] where \(n! = n(n-1)(n-2) \cdots 1\) for \(n \geq 1\) and \(0! = 1.\)

Any trial or experiment whose outcome can be classified as either a success or a failure is called a **Bernoulli trial**.

If the probability of success is \(p,\) the probability of failure is \( 1-p,\) and when \( T \) consecutive trials are performed, the number of successes \( N\) is distributed as
\[ P(N=n) = {T \choose n} p^{n} (1-p)^{T-n};\]
this distribution for \( N \) is called **binomial** because of the binomial coefficient \({T \choose n}.\)

The coin flip is an example with \( p = \frac{1}{2}, T = 10\); we'll see more examples later in the course.

Once we have \(X\)'s probability distribution, we can compute the **expectation value** \(E[X],\) which is the sum of all of \(X\)'s values weighted by their likelihoods:
\[E[X] = \sum\limits_{n \in X\text{'s range}} n P(X = n).\]
Let's say an *unfair* coin comes up heads three times out every four flips. We flip this coin twice in a row; if \(N\) is the number of times it lands heads up, \(N\)'s probability distribution is
\[ P(N = n ) = {2 \choose n}\left( \frac{3}{4} \right)^{n} \left( \frac{1}{4} \right)^{2-n}.\]
What's the average number of heads we should expect to see if we perform this coin flip experiment many times?

**Good to Know:** \( \sum\limits_{j=0}^{n} a_{j} \) means \( a_{0} + a_{1} + \dots + a_{n} \) and
\[ {2 \choose 0} = 1, \ {2 \choose 1 } = 2 \ , {2 \choose 2 } = 1.\]

Another valuable quantity we can find from \( X\)'s distribution is its **variance** \[\begin{align} \text{Var}[X]& = E[(X-E[X])^2]= E[X^2- 2 E[X] X +(E[X])^2] \\ & = E[X^2]-2 (E[X])^2+(E[X])^2 \\ & = E\big[X^2\big] - \big( E[X]\big)^2,\end{align}\] which measures the spread of observed values away from the expected one.

What's the spread of the double-flip coin experiment from the last problem? Remember, if \(N\) is the number of heads, \[ P(N = n ) = {2 \choose n}\left( \frac{3}{4} \right)^{n} \left( \frac{1}{4} \right)^{2-n}\ \ \text{where} \ {2 \choose 0} = 1, \ {2 \choose 1 } = 2 \ , {2 \choose 2 } = 1.\]

We find it useful to assign distribution functions to continuous random variables as well: \[ P( a \leq X \leq b) = \int\limits_{x = a }^{x=b} f(x) dx \ \small{\text{for some } \textbf{density function }} f(x). \] The integral means the area below \(f\)'s graph and above the \(x\)-axis between \( a \) and \(b.\) In a loose sense, we think of \(\int\) as a sum of continuous values, much like \( \sum\) is a sum of discrete ones.

For instance, if the sample space \( \Omega \) is a fixed interval, say \( [-2,3],\) we can choose \( f(x) \) to be a constant; this choice gives us the **uniform distribution**.

What constant value must \( f \) have in order for this density to define a probability distribution on \( \Omega = [-2,3] ? \)

**Hint:** In each random experiment, it's certainly the case that we draw at least *some* \( x \) from \( \Omega.\)

The density function allows us to find interesting characteristics of a continuous random variable, like its **expected value**:
\[ E[X] = \int\limits_{\text{range of } X} x f(x)\, dx, \]
which is very similar to the version for discrete random variables.

What is the expected value of the uniform distribution \( f = \frac{1}{5} \) on \( \Omega = [-2,3] ?\)

**Hint:** If you want to avoid using calculus, remember that the graph of \( \frac{x}{5} \) is a straight line through the origin, so \( E[X]\) is related to the area of two triangles, one on either side of \( x= 0.\)

What is the **variance** of the uniform distribution on \( \Omega = [ -2,3] ?\) Recall that
\[ \text{Var}[X] = E\big[X^2\big] - \big( E[X]\big)^2.\]

**Good to Know:** If you want to avoid calculus, the area below the parabola \(y=x^2\) and above the \(x\)-axis between \( a\) and \( b\) is \( \frac{1}{3} [b^3-a^3].\)

Discrete and continuous **random variables** and **probability distributions** are the central characters in our course.

We introduced these ideas in this quiz and the last; in the next quiz, we'll review by applying them to a few interesting real-world problems.

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