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Introduction to Probability and Statistics for the Life Sciences Spring 2026 MATH 1044

Andy Eisenberg

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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.
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Show all work unless instructed otherwise.
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1.

Write either True or False for each of the following statements. No justification is required.
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(b)

Suppose \(X\) is a continuous random variable with pdf \(f(x)\) and cdf \(F(x)\text{.}\) Then \(\displaystyle{\int_a^b f(x)\ dx = F(b) - F(a)}\text{.}\)
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Solution.
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2.

Consider the joint distribution for \(X\) and \(Y\) below.
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Table 116. Joint Distribution
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\(X = 0\) \(X = 1\) \(X = 2\)
\(Y = 0\) 0.1 0.05 0.1
\(Y = 1\) 0.3 0.15 0.3

(a)

Find the marginal distributions for \(X\) and \(Y\text{.}\)
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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*}
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(b)

Are \(X\) and \(Y\) independent?
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Solution.
Checking each cell in the table:
\begin{align*} \Pr(X = 0, Y = 0) \amp = 0.1 = (0.4)(0.25) = \Pr(X = 0)\Pr(Y = 0) \\ \Pr(X = 1, Y = 0) \amp = 0.05 = (0.2)(0.25) = \Pr(X = 1)\Pr(Y = 0) \\ \Pr(X = 2, Y = 0) \amp = 0.1 = (0.4)(0.25) = \Pr(X = 2)\Pr(Y = 0) \\ \Pr(X = 0, Y = 1) \amp = 0.3 = (0.4)(0.75) = \Pr(X = 0)\Pr(Y = 1) \\ \Pr(X = 1, Y = 1) \amp = 0.15 = (0.2)(0.75) = \Pr(X = 1)\Pr(Y = 1) \\ \Pr(X = 2, Y = 1) \amp = 0.3 = (0.4)(0.75) = \Pr(X = 2)\Pr(Y = 1) \end{align*}
So \(X, Y\) are independent.
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3.

Suppose a continuous random variable \(X\) taking values in \([0, 1]\) has cdf \(F(x) = 2x^2 - x^4\text{.}\)
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(b)

Find \(\E(X)\text{.}\)
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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^3}{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*}
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