Discrete CDF to PMF

1. The discrete random variable \(X\) has probability function \(\mathrm { p } ( x )\) and cumulative distribution function \(\mathrm { F } ( x )\) given in the table below.

1. Write down the value of \(d\)
2. Find the values of \(a\), \(b\) and \(c\)
3. Write down the value of \(\mathrm { P } ( X > 4 )\) Two independent observations, \(X _ { 1 }\) and \(X _ { 2 }\), are taken from the distribution of \(X\).
4. Find the probability that \(X _ { 1 }\) and \(X _ { 2 }\) are both odd. Given that \(X _ { 1 }\) and \(X _ { 2 }\) are both odd,
5. find the probability that the sum of \(X _ { 1 }\) and \(X _ { 2 }\) is 6 Give your answer to 3 significant figures.

1. A biased four-sided die has faces marked \(1,3,5\) and 7 . The random variable \(X\) represents the score on the die when it is rolled. The cumulative distribution function of \(X , \mathrm {~F} ( x )\), is given in the table below.

1. Find the probability distribution of \(X\)
2. Find \(\mathrm { P } ( 2 < X \leqslant 6 )\)
3. Write down the value of \(\mathrm { F } ( 4 )\)

2. The discrete random variable \(X\) can take only the values 1,2 and 3 . For these values the cumulative distribution function is defined by $$\mathrm { F } ( x ) = \frac { x ^ { 3 } + k } { 40 } \quad x = 1,2,3$$

1. Show that \(k = 13\)
2. Find the probability distribution of \(X\). Given that \(\operatorname { Var } ( X ) = \frac { 259 } { 320 }\)
3. find the exact value of \(\operatorname { Var } ( 4 X - 5 )\).

4. A discrete random variable \(X\) takes only positive integer values. It has a cumulative distribution function \(\mathrm { F } ( x ) = \mathrm { P } ( X \leq x )\) defined in the table below.

1. Determine the probability function, \(\mathrm { P } ( X = x )\), of \(X\).
2. Calculate \(\mathrm { E } ( X )\) and show that \(\operatorname { Var } ( X ) = 5.76\).
3. Given that \(Y = 2 X + 3\), find the mean and variance of \(Y\).

6. The discrete random variable \(X\) can take only the values 2,3 or 4 . For these values the cumulative distribution function is defined by $$F ( x ) = \frac { ( x + k ) ^ { 2 } } { 25 } \text { for } x = 2,3,4$$ where \(k\) is a positive integer.

1. Find \(k\).
2. Find the probability distribution of \(X\).

2．The discrete random variable \(X\) takes the values 1,2 and 3 and has cum
function \(\mathrm { F } ( x )\) given by \includegraphics[max width=\textwidth, alt={}, center]{4cf4f2d7-d912-4b65-a666-caa37009661a-04_24_37_182_2010}

4. The discrete random variable \(X\) has probability distribution The cumulative distribution function of \(X\) is given by

1. Find the values of \(a , b , c , d\) and \(e\).
2. Write down the value of \(\mathrm { P } \left( X ^ { 2 } = 1 \right)\).
   \section*{} \section*{
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   } \(T\)

1. The discrete random variable \(X\) has the following probability distribution

where \(a , b\) and \(c\) are probabilities.
The cumulative distribution function of \(X\) is \(\mathrm { F } ( x )\) and \(\mathrm { F } ( 3 ) = 0.2\) and \(\mathrm { F } ( 6 ) = 0.8\)

1. Find the value of \(a\), the value of \(b\) and the value of \(c\).
2. Write down the value of \(\mathrm { F } ( 7 )\).

1 The discrete random variable \(A\) takes only the values 0,2 and 4, and has cumulative distribution function \(\mathrm { F } ( a ) = \mathrm { P } ( A \leq a )\) Find \(\mathrm { P } ( A = 2 )\)
Circle your answer. \(0 \quad 0.4 \quad 0.6 \quad 0.8\)