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. \(0 \quad 0.4 \quad 0.6 \quad 0.8\)

The discrete random variable \(Y\) has probability distribution where \(a\), \(b\) and \(c\) are constants. The cumulative distribution function F(\(y\)) of \(Y\) is given in the following table where \(d\) is a constant.

1. Find the value of \(a\), the value of \(b\), the value of \(c\) and the value of \(d\). [5]
2. Find \(\text{P}(3Y + 2 \geq 8)\). [2]