Discrete probability distributionsEdexcel A-Level Maths: Revision notes

A discrete random variable $X$ takes separate values $x$, each with a probability $P(X=x)$. A probability distribution lists these in a table or gives a formula. Two rules always hold: every $P(X=x)$ lies between $0$ and $1$, and $\sum P(X=x)=1.$ Because the values are mutually exclusive, the probability of several values is found by adding: $P(X\ge3)=P(X=3)+P(X=4)$. Example: $P(X=x)=kx$ for $x=1,2,3,4$ gives $10k=1$, $k=\frac1{10}$ and $P(X\ge3)=0.7$. (Calculating the mean and variance is not needed here.)

Set the sum of probabilities equal to $1$ and solve. Example: $P(Y=0)=0.15$, $P(Y=1)=a$, $P(Y=2)=2a$, $P(Y=3)=a+0.05$. Then $0.2+4a=1$, so $a=0.2$ and the distribution is $0.15,\ 0.2,\ 0.4,\ 0.25$. Check that every value is between $0$ and $1$ and that they total $1$. A solution giving a negative probability must be rejected.

A discrete uniform distribution has $n$ values, all equally likely, so $P(X=x)=\frac1n$ for each. Examples: a fair die ($n=6$) or a fair eight-sided die, with $P(X=x)=\frac18$. Probabilities of events are then counted: $P(X>5)=\frac38$ for the eight-sided die, and $P(X\text{ prime})=\frac48=\frac12$. To identify one, check that all the values have the same probability. The total of two spins is not uniform, because some totals can be made in more ways than others.

If two values of $X$ are observed independently, then $P(X_1=a\text{ and }X_2=b)=P(X=a)P(X=b)$. To find the probability of a total, list every pair that gives it and add the products. With $P(X=x)=0.1x$: a total of $3$ needs $(1,2)$ or $(2,1)$, so $2\times0.1\times0.2=0.04$. For two rolls of a fair eight-sided die, $P(\text{first}>\text{second})=\frac12\left(1-\frac18\right)=\frac7{16}$ by symmetry. Remember to count both orders when the values are different.