SACE Stage 2 Maths Methods Topic Tests
Discrete Random Variables Topic Test 1
Number of marks: 10
Writing time: 13 minutes
Instructions
• Show appropriate working and steps of logic in the question booklets
• State all answers correct to three significant figures, unless otherwise instructed
• Use black or blue pen
• You may use a sharp dark pencil for diagrams and graphical representations
Consider a discrete random variable, \(X\), which has the probability mass function \[ \Pr(X=x) = \frac{e^{-2}2^x}{x!} \] where \(x\) is a non-negative integer, and \(x! = x \times (x-1) \times (x-2) \times \dots \times 3 \times 2 \times 1\) with \(0!=1\).
(a) Based on the information provided, complete the table below by calculating the missing probabilities. State your answers correct to four decimal places. (2 marks)
| \(\Pr(X=0)\) | \(\Pr(X=1)\) | \(\Pr(X=2)\) | \(\Pr(X=3)\) | \(\Pr(X=4)\) | \(\Pr(X=5)\) | \(\Pr(X \ge 6)\) |
| 0.2707 | 0.2707 | 0.1804 | 0.0902 | 0.0166 |
A moment-generating function can be used to find key properties of a distribution without using the probability mass function directly. For the probability mass function given above, the moment-generating function is \[ M(t) = e^{2e^t-2}. \]
(b) The mean, \(\mu\), of the probability mass function given above can be found using the formula \[ \mu = M'(0). \] Show that \(\mu=2\). (2 marks)
(c) (i) Show that \(M''(t) = 2e^{2e^t-2}(2e^{2t}+e^t)\). (2 marks)
(ii) The standard deviation, \(\sigma\), of the probability mass function given above can be found using the formula \[ \sigma = \sqrt{M''(0) - \mu^2}. \] Show that \(\sigma = \sqrt{2}\). (2 marks)
(d) Hence, determine the probability that \(X\) is within one standard deviation of the mean. (2 marks)
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