Discrete Random Variable Probability Distribution

00:00 14 marks

A discrete random variable \(X\) has the following probability distribution:

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

(a) Find the value of \(k\). [3]
(b) Find \(E(X)\). [3]

Consider the events \(A\) where \(X\) is odd and \(B\) where \(X > 1\).
(c) Find \(P(A)\). [2]
(d) Find \(P(B)\). [2]
(e) Find \(P(A \cup B)\). [2]
(f) Find \(P(A | B)\). [2]

Part a

3 minutes 3 marks

Question

Find the value of \(k\). [3]

Hint Hint

Hint

Sum the probabilities: \(k + 2k + 3k + 4k = 1\), then solve for \(k\).

Answer Answer
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Complete Answer

k = 1/10

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Part b

3 minutes 3 marks

Question

Find \(E(X)\). [3]

Hint Hint

Hint

Use the formula \(E(X)=\sum x \cdot P(X=x)\) and substitute \(k = \frac{1}{10}\).

Answer Answer
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Complete Answer

E(X) = 2

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Part c

2 minutes 2 marks

Question

Find \(P(A)\), where event A is \(X\) being odd. [2]

Hint Hint

Hint

Event A includes outcomes \(X = 1\) and \(X = 3\). Add their probabilities.

Answer Answer
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Complete Answer

P(A) = 3/5

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Part d

2 minutes 2 marks

Question

Find \(P(B)\), where event B is \(X > 1\). [2]

Hint Hint

Hint

Event B includes outcomes \(X = 2\) and \(X = 3\). Sum their probabilities.

Answer Answer
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Complete Answer

P(B) = 7/10

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Part e

2 minutes 2 marks

Question

Find \(P(A \cup B)\). [2]

Hint Hint

Hint

Since \(A \cup B\) includes outcomes where \(X=1,2,3\), sum the corresponding probabilities.

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Complete Answer

P(A ∪ B) = 9/10

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Part f

2 minutes 2 marks

Question

Find \(P(A | B)\). [2]

Hint Hint

Hint

Recall \(P(A | B)=\frac{P(A \cap B)}{P(B)}\). Note that \(A \cap B\) is only \(X=3\).

Answer Answer
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Complete Answer

P(A | B) = 4/7

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