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Worksheet Quiz 2
The following work should be completed individually. Use of notes or textbooks is not allowed. You may use a scientific calculator, not a graphing calculator or phone app.
Show all work unless instructed otherwise.
1. Write either True or False for each of the following statements. No justification is required.
(a) Let
\(n, k\) be integers with
\(0 \leq k \leq n\text{.}\) Then
\({n\choose k} = {n \choose n - k}\text{.}\)
(b) Suppose
\(X\) is a continuous random variable with pdf
\(f(x)\) and cdf
\(F(x)\text{.}\) Then
\(\displaystyle{\int_a^b f(x)\ dx = F(b) - F(a)}\text{.}\)
(c) Suppose
\(X\) is a random variable taking values between 0 and 6. Then
\(\E(X) = 3\text{.}\)
2.
Consider the joint distribution for
\(X\) and
\(Y\) below.
Table 90. Joint Distribution
\(X = 0\) \(X = 1\) \(X = 2\)
\(Y = 0\) 0.1
0.05
0.1
\(Y = 1\) 0.3
0.15
0.3
(a) Find the marginal distributions for
\(X\) and
\(Y\text{.}\)
Solution .
\begin{align*}
\Pr(X = 0) \amp = 0.1 + 0.3 = 0.4 \amp \Pr(Y = 0) \amp = 0.1 + 0.05 + 0.1 = 0.25 \\
\Pr(X = 1) \amp = 0.05 + 0.15 = 0.2 \amp \Pr(Y = 1) \amp = 0.3 + 0.15 + 0.3 = 0.75 \\
\Pr(X = 2) \amp = 0.1 + 0.3 = 0.4
\end{align*}
(b) Are
\(X\) and
\(Y\) independent?
Solution .
\(\Pr(X = 2, Y = 0) = 0.2 \text{,}\) and
\(\Pr(X = 2)\Pr(Y = 0) = (0.4)(0.25) = 0.1\text{.}\) Since
\(0.2 \neq 0.1\text{,}\) \(X\) and
\(Y\) are not independent.
3. Suppose a continuous random variable
\(X\) taking values in
\([0, 1]\) has cdf
\(F(x) = 2x^2 - x^4\text{.}\)
(a) Find the pdf
\(f(x)\text{.}\)
Solution .
\begin{gather*}
f(x) = F'(x) = 4x - 4x^3.
\end{gather*}
(b) Solution .
\begin{align*}
\E(X) \amp = \int_0^1 x f(x)\ dx \\
\amp = \int_0^1 x (4x - 4x^3)\ dx \\
\amp = \int_0^1 4x^2 - 4x^4\ dx \\
\amp = \frac{4x^2}{3} - \frac{4x^5}{5}\bigg|_0^1 \\
\amp = \left(\frac{4}{3} - \frac{4}{5}\right) - (0) \\
\amp = \frac{8}{15} \approx 0.533.
\end{align*}
