Find parameter n from mean

3 The random variable \(X\) has a discrete uniform distribution and takes values \(1,2,3 , \ldots , n\) The mean of \(X\) is 8 3

1. Show that \(n = 15\) [0pt] [2 marks]
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2. \(\quad\) Find \(\mathrm { P } ( X > 4 )\) 3
3. Find the variance of \(X\), giving your answer in exact form.

5. The random variable \(X\) has the discrete uniform distribution $$\mathrm { P } ( X = x ) = \frac { 1 } { n } , \quad x = 1,2 , \ldots , n$$ Given that \(\mathrm { E } ( X ) = 5\),

1. show that \(n = 9\). Find
2. \(\mathrm { P } ( X < 7 )\),
3. \(\operatorname { Var } ( X )\).

The random variable \(X\), which can take any value from \(\{1, 2, \ldots, n\}\), is modelled by the discrete uniform distribution with mean 10.

1. Show that \(n = 19\) and find the variance of \(X\). [4 marks]
2. Find \(\text{P}(3 < X \leq 6)\). [2 marks]

The random variable \(Y\) is defined by \(Y = 3(X - 10)\).

1. State the mean and the variance of \(Y\). [3 marks]

The model for the distribution of \(X\) is found to be unsatisfactory, and in a refined model the probability distribution of \(X\) is taken to be $$\text{f}(x) = \begin{cases} k(x + 1) & x = 1, 2, \ldots, 19, \\ 0 & \text{otherwise}. \end{cases}$$

1. Show that \(k = \frac{1}{209}\). [3 marks]
2. Find \(\text{P}(3 < X \leq 6)\) using this model. [3 marks]