Expected Value and Variance

STAT 20: Introduction to Probability and Statistics

Some thoughts on expected values…

Concept Review

Let \(X\) be a random variable such that \[ X = \begin{cases} -1, & \text{ with probability } 1/3\\ 0, & \text{ with probability } 1/6\\ 1, & \text{ with probability } 4/15 \\ 2, & \text{ with probability } 7/30 \\ \end{cases} \]

1. Draw the graph of the cdf of \(X\)

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2. Compute the expected value and variance of \(X\)

write down pmf with denominator 30, and draw cdf on the board. P(X = -1)= 5/30, P(X = 0)= 10/30, P(X = 1)= 8/30, P(X = 2)= 7/30 E(X) = (-1)(5/30) + 0(10/30) + 1(8/30) + 2(7/30) (-10 + 0 + 8 + 14)/30 = 12/30 = 2/5 Var(X) = E(X^2) - 4/25 = (10 + 0 + 8 +28)/30 -4/25 = 23/15- 4/25 ~~ 1.37333 Graph the pmf and mark the expectation The numbers are not pretty but the point is to show them the computation and graph The rest of the ideas can be recapped as you go over the concept questions You may want to review linearity of expectation here, or save it for when going over the PS

Concept Questions

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\(X\) is a random variable with the distribution shown below:

\[ X = \begin{cases} 3, \; \text{ with prob } 1/3\\ 4, \; \text{ with prob } 1/4\\ 5, \; \text{ with prob } 5/12 \end{cases} \]

Consider the box with tickets: \(\fbox{3}\, \fbox{3}\, \fbox{3} \,\fbox{4} \,\fbox{4} \,\fbox{4} \,\fbox{4} \,\fbox{5} \,\fbox{5}\, \fbox{5} \,\fbox{5} \,\fbox{5}\)

Suppose we draw once from this box and let \(Y\) be the value of the ticket drawn. Which random variable has a higher expectation?

The expected value of \(X\) is ____ the expected value of \(Y\).

This is a reading question. Note that the pmf implied by the box is not the same as the given pmf. In the box, 4 has a higher probability, so the average is higher. They do not need to actually do the computation. At this point, ask them what box would they use so that if \(X\) is the value of a randomly selected ticket, the distribution of \(X\) is exactly the distribution shown here.

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A prof notices that Zoom office hours are not too crowded this semester. She observes that number of Stat 20 students coming to her Thursday office hours have a Poisson(2) distribution. There is one Data 88 student from a previous semester who is always there (they want a letter of recommendation).

What is the expected value (EV) and variance (V) of the number of students in her Zoom office hours?

This is also a reading question, with < 50% getting it right. Let \(X\) be the number of Stat 20 students who go to the office hours. Then \(E(X) = 2 = Var(X)\). But the number of students in the office is \(X+1\) since that one student is always there. \(E(X+1) = 2+1 = 3\) and \(Var(X+1) = Var(X) = 2\)

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Let \(X\) be a discrete uniform random variable on the set \(\{-1, 0, 1\}\).

If \(Y=X^2\), what is \(E(Y)\)?

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Let \(X\) be a discrete uniform random variable on the set \(\{-1, 0, 1\}\).

If \(W = \min(X, 0.5)\), what is \(E(W)\)?

begin by asking them what is What is \(E(X)\)? And pointing out that if we have a symmetric distribution then EX is in the middle. Emphasize the idea of it being a “typical” value and an average or a mean. This is so they practice computing the EV of a function of a rv. Have them make a table (or perhaps you draw a table) of \(X\), \(Y\), \(W\) and \(f(x)\)