document.write( "Question 1192779: The sample space of a random experiment is {a, b, c, d, e, f}, and each outcome is equally likely. A random variable is defined as follows
\n" ); document.write( "Outcome a b c d e f
\n" ); document.write( "X 0 0 1.5 1.5 2 3\r
\n" ); document.write( "\n" ); document.write( "Determine the probability mass function of X.
\n" ); document.write( " \n" ); document.write( "
Algebra.Com's Answer #848951 by CPhill(1959)\"\" \"About 
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The **probability mass function (PMF)** of a random variable \( X \) provides the probability of each possible value of \( X \). Here's how we determine the PMF:\r
\n" ); document.write( "\n" ); document.write( "### Step 1: Analyze the sample space and probabilities
\n" ); document.write( "The sample space is \( S = \{a, b, c, d, e, f\} \), and each outcome is equally likely. Since there are 6 outcomes:
\n" ); document.write( "\[
\n" ); document.write( "P(\text{each outcome}) = \frac{1}{6}.
\n" ); document.write( "\]\r
\n" ); document.write( "\n" ); document.write( "### Step 2: Group outcomes by the values of \( X \)
\n" ); document.write( "The random variable \( X \) assigns values to each outcome as follows:
\n" ); document.write( "- \( X(a) = 0 \),
\n" ); document.write( "- \( X(b) = 0 \),
\n" ); document.write( "- \( X(c) = 1.5 \),
\n" ); document.write( "- \( X(d) = 1.5 \),
\n" ); document.write( "- \( X(e) = 2 \),
\n" ); document.write( "- \( X(f) = 3 \).\r
\n" ); document.write( "\n" ); document.write( "We can group the outcomes by the values of \( X \):
\n" ); document.write( "- \( X = 0 \): Outcomes \( a, b \) → Probability = \( P(X = 0) = P(a) + P(b) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \).
\n" ); document.write( "- \( X = 1.5 \): Outcomes \( c, d \) → Probability = \( P(X = 1.5) = P(c) + P(d) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3} \).
\n" ); document.write( "- \( X = 2 \): Outcome \( e \) → Probability = \( P(X = 2) = P(e) = \frac{1}{6} \).
\n" ); document.write( "- \( X = 3 \): Outcome \( f \) → Probability = \( P(X = 3) = P(f) = \frac{1}{6} \).\r
\n" ); document.write( "\n" ); document.write( "### Step 3: Write the PMF
\n" ); document.write( "The PMF of \( X \) is:
\n" ); document.write( "\[
\n" ); document.write( "P(X = x) =
\n" ); document.write( "\begin{cases}
\n" ); document.write( "\frac{1}{3} & \text{if } x = 0, \\
\n" ); document.write( "\frac{1}{3} & \text{if } x = 1.5, \\
\n" ); document.write( "\frac{1}{6} & \text{if } x = 2, \\
\n" ); document.write( "\frac{1}{6} & \text{if } x = 3, \\
\n" ); document.write( "0 & \text{otherwise.}
\n" ); document.write( "\end{cases}
\n" ); document.write( "\] \n" ); document.write( "
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