Discrete Random Variable Probability Distribution

00:00 14 marks

11. [Maximum mark: 14]
A discrete random variable \(X\) has the following probability distribution:

\(x\)	0	1	2	3
\(P(X=x)\)	\(k\)	2\(k\)	3\(k\)	4\(k\)

(a) Find the value of \(k\).
(b) Find \(E(X)\).

Consider the events \(A\) where \(X\) is odd and \(B\) where \(X > 1\).
(c) Find \(P(A)\).
(d) Find \(P(B)\).
(e) Find \(P(A \cup B)\).
(f) Find \(P(A \mid B)\).

Part a

2 minutes 3 marks

Question

Find the value of \(k\).

Hint Hint

Hint

Set up the equation \(k + 2k + 3k + 4k = 10k = 1\) and solve for k.

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Complete Answer

\(k = \frac{1}{10}\)

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Part b

2 minutes 3 marks

Question

Find \(E(X)\).

Hint Hint

Hint

Compute \(E(X)=0\cdot k + 1\cdot (2k) + 2\cdot (3k) + 3\cdot (4k)\) and substitute the value of k.

Answer Answer
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Complete Answer

\(E(X)=20k=20/10=2\)

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Part c

1 minutes 2 marks

Question

Find \(P(A)\) where event \(A\) is defined as \(X\) being odd.

Hint Hint

Hint

Odd outcomes are when \(X=1\) and \(X=3\).

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Complete Answer

\(P(A)=2k+4k=6k=\frac{3}{5}\)

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Part d

1 minutes 2 marks

Question

Find \(P(B)\) where event \(B\) is defined as \(X > 1\).

Hint Hint

Hint

For \(X > 1\), consider \(X=2\) and \(X=3\).

Answer Answer
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\(P(B)=3k+4k=7k=\frac{7}{10}\)

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Part e

1 minutes 2 marks

Question

Find \(P(A \cup B)\).

Hint Hint

Hint

Use the formula \(P(A \cup B)=P(A)+P(B)-P(A \cap B)\). Note that \(A \cap B\) is when \(X=3\).

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Complete Answer

\(P(A \cup B)=6k+7k-4k=9k=\frac{9}{10}\)

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Part f

1 minutes 2 marks

Question

Find \(P(A \mid B)\).

Hint Hint

Hint

Use the conditional probability formula: \(P(A \mid B)=\frac{P(A \cap B)}{P(B)}\).

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Complete Answer

\(P(A \mid B)=\frac{4k}{7k}=\frac{4}{7}\)

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