# Variance
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## Section 3.2 Variance

Text of section.
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### Exercises Exercises

#### 1.

Consider a random variable \(X\) with probability distribution below. Find \(\Var(X)\text{.}\)
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Table 3.2.1.
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\(x\) | \(\Pr(X = x)\)
--- | ---
1 | 0.1
2 | 0.05
3 | 0.2
4 | 0.15
5 | 0.15
6 | 0.1
7 | 0.05
8 | 0.1
9 | 0.1

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#### 2.

Let \(X\) be a random variable taking the values 1, 2, 3, 4, 5. Write a distribution table for \(X\text{,}\) then use your table to write a distribution for \(X^2\text{.}\) Then, find \(\Var(X)\text{.}\)
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#### 3.

Suppose we flip a coin \(n = 100\) times, and let \(N\) count the number of heads. If the coin comes up heads on a flip with probability \(p = 0.4\text{,}\) what is \(\Var(N)\text{?}\) What if \(n = 80\) and \(p = 0.6\text{?}\) What if \(n = 200\) and \(p = 0.5\text{?}\)
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#### 4.

If \(\E(X) = 3\text{,}\) \(\Var(X) = 2\text{,}\) what is \(\E(X^2)\text{?}\)
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#### 5.

A continuous random variable \(X\) taking values in \([0, 1]\) has p.d.f. \(f(x) = 2x\text{.}\) What is \(\Var(X)\text{?}\)
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#### 6.

A continuous random variable \(X\) taking values in \([-1, 1]\) has p.d.f. \(f(x) = \frac{3x^2}{2}\text{.}\) What is \(\Var(X)\text{?}\)
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#### 7.

A continuous random variable \(X\) taking values in \([1, 4]\) has p.d.f. \(f(x) = \frac{4}{3x^2}\text{.}\) What is \(\Var(X)\text{?}\)
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#### 8.

A continuous random variable \(X\) taking values in \([1, 4]\) has p.d.f. \(f(x) = k(x - \sqrt{x})\) for some constant \(k\text{.}\) In a previous problem, you found the value of \(k\text{.}\) Now, find \(\Var(X)\text{.}\)
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#### 9.

A continuous random variable \(X\) taking values in \([1, 2]\) has p.d.f. \(\displaystyle{f(x) = \frac{1}{2}\left(\frac{1}{x^2} + x\right)}\text{.}\) Find \(\Var(X)\text{.}\)
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