The random variable $X$ has a discrete distribution. The probability that it takes a value other than 13, 14, 15 or 16 is negligible. (a) Complete the probability distribution table below and hence calculate $E(X)$, the expected value of $X$. | $x$ | 13 | 14 | 15 | 16 | |----------|---------|---------|---------|---------| | $P(X = x)$ | 0.383 | 0.575 | 0.004 | (b) If $X$ is the age, in complete years, on 1 January 2013 of a student selected at random from among all second-year students in Irish schools, explain what $E(X)$ represents. (c) If ten students are selected at random from this population, find the probability that exactly six of them were 14 years old on 1 January 2013. Give your answer correct to three significant figures.

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The random variable $X$ has a discrete distribution - Leaving Cert Mathematics - Question 1 - 2014

Question 1

The random variable $X$ has a discrete distribution. The probability that it takes a value other than 13, 14, 15 or 16 is negligible. (a) Complete the probability d... show full transcript

Worked Solution & Example Answer:The random variable $X$ has a discrete distribution - Leaving Cert Mathematics - Question 1 - 2014

Step 1

Complete the probability distribution table and calculate $E(X)$

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Answer

To complete the probability distribution table, we need to ensure that the sum of the probabilities equals 1. We can find $P(X = 15)$ as follows:

$P(15) = 1 - (P(13) + P(14) + P(16)) = 1 - (0.383 + 0.575 + 0.004) = 1 - 0.962 = 0.038$

Now, we can calculate the expected value $E(X)$ :

$E(X) = ext{sum}(x imes P(X = x)) = 13 imes 0.383 + 14 imes 0.575 + 15 imes 0.038 + 16 imes 0.004$ $E(X) = 4.979 + 8.050 + 0.570 + 0.064 = 13.663$

Step 2

Explain what $E(X)$ represents.

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Answer

$E(X)$ represents the mean of the ages of all second-year students in Irish schools on 1 January 2013. It provides a measure of central tendency, giving an indication of the average age in this population.

Step 3

Find the probability that exactly six of them were 14 years old.

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Answer

We can model this using the binomial distribution. Let $p = P(X = 14) = 0.575$ . The number of trials $n = 10$ , and we want exactly $k = 6$ successes:

$P(X = 6) = {10 \\choose 6} p^6 (1 - p)^{10 - 6}$ Substituting in the values: $P(X = 6) = {10 \\choose 6} (0.575)^6 (0.425)^4$ Calculating: $= 210 imes (0.575)^6 imes (0.425)^4 \\approx 0.248$ Thus, the probability that exactly six of the ten students are 14 years old is approximately 0.248.

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The random variable $X$ has a discrete distribution - Leaving Cert Mathematics - Question 1 - 2014

Worked Solution & Example Answer:The random variable $X$ has a discrete distribution - Leaving Cert Mathematics - Question 1 - 2014

Complete the probability distribution table and calculate $E(X)$

Explain what $E(X)$ represents.

Find the probability that exactly six of them were 14 years old.

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