Define discrete probability distribution .

A discrete probability distribution describes all the values a discrete random variable can take, together with their probabilities . It can be given in three ways, and questions move between them freely: as a table , with the values along the top and the probabilities underneath as a function , called a probability mass function as a vertical line graph , with the values along the horizontal axis and the probabilities as the heights of the lines

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Probability Distributions (Edexcel A Level Maths: Statistics): Flashcards

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FrontDiscrete Probability Distributions
Define discrete random variable.
BackDiscrete Probability Distributions
A random variable is a variable whose value depends on the outcome of a random event, so it is not known until the event happens.
It is discrete if it can take only certain separate values rather than any value in a range, so a discrete variable usually counts something, like the number of heads in 20 coin flips.
The values themselves need not be whole numbers: a spinner with $\frac{1}{3}$ on one of its sectors still gives a discrete random variable, because the values it can take are separate.

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Define discrete random variable.
A random variable is a variable whose value depends on the outcome of a random event, so it is not known until the event happens.
It is discrete if it can take only certain separate values rather than any value in a range, so a discrete variable usually counts something, like the number of heads in 20 coin flips.
The values themselves need not be whole numbers: a spinner with $\frac{1}{3}$ on one of its sectors still gives a discrete random variable, because the values it can take are separate.
Why is $X$ written as a capital letter and $x$ as a small one?
They mean different things, and the convention is used throughout the rest of the course.
The capital letter is the random variable itself, the quantity whose value is not yet known, and the small letter is a particular value that the variable might take.
So $P (X = x)$ reads "the probability that the random variable $X$ takes the value $x$ ".
True or False?
A discrete random variable can only take a finite number of values.
False.
Usually it can, but not always. What makes a variable discrete is that its values are separate, not that there are finitely many of them.
So, for example, the number of emails a manager receives in an hour, and the number of times a dice is rolled until it lands on a 6, can both be $1, 2, 3, \dots$ with no upper limit.
Define discrete probability distribution.
A discrete probability distribution describes all the values a discrete random variable can take, together with their probabilities.
It can be given in three ways, and questions move between them freely:
- as a table, with the values along the top and the probabilities underneath
- as a function, called a probability mass function
- as a vertical line graph, with the values along the horizontal axis and the probabilities as the heights of the lines
Complete the property that every discrete probability distribution has:
$Σ P (X = x) =______$
The completed property is:
$Σ P (X = x) = 1$
The variable is certain to take one of its values, so the probabilities of all of them must account for the whole of the probability.
This is what lets you find an unknown: if $P (X = x) = k x^{2}$ for $x = - 3, - 1, 2, 4$ , then $9 k + k + 4 k + 16 k = 1$ , giving $30 k = 1$ and $k = \frac{1}{30}$ .
It is also a check: every probability must lie between 0 and 1, and they must total 1.
What is a discrete uniform distribution?
A discrete uniform distribution is one where the random variable takes a finite number of values, each with an equal probability.
If there are $n$ values then each one has probability $\frac{1}{n}$ .
So, for example, the score on a fair six-sided dice is discrete uniform, with each of the six scores having probability $\frac{1}{6}$ , while a spinner with unequal sectors is not.
How do you find $P (X \leq k)$ from a discrete probability distribution?
Identify every value the variable can take that satisfies the inequality, then add their probabilities:
$P (X \leq k) = \underset{x_{i} \leq k}{Σ} P (X = x_{i})$
The value $k$ itself need not be one of them: if $X$ can only be $- 3, - 1, 2$ or 4, then $P (X \leq 3)$ adds the first three, since 3 is not a possible value.
The cumulative probability is sometimes written $F (x) = P (X \leq x)$ , and it is often quicker to subtract the unwanted values from 1 than to add the wanted ones.
Complete the inequality that each phrase translates into, for a discrete random variable $X$ :
"at most $k$ " becomes $X______k$
"fewer than $k$ " becomes $X______k$
"at least $k$ " becomes $X______k$
The completed inequalities are:
"at most $k$ " becomes $X \leq k$
"fewer than $k$ " becomes $X < k$
"at least $k$ " becomes $X \geq k$
"No greater than" behaves like "at most", and "no fewer than" like "at least". "Greater than $k$ " gives $X > k$ .
The whole answer turns on whether $k$ itself is included, so it is worth reading these phrases slowly.
$X$ can take the values $- 3, - 1, 2$ and 4. How do you find $P (X^{2} < 5)$ ?
Exactly as for any other event: work out which of the possible values satisfy it, then add their probabilities.
Testing each one, ${(- 3)}^{2} = 9$ and $4^{2} = 16$ are both too big, while ${(- 1)}^{2} = 1$ and $2^{2} = 4$ are not, so
$P (X^{2} < 5) = P (X = - 1) + P (X = 2)$
Do not try to rearrange the inequality into one about $X$ and read it off: squaring makes $- 3$ behave like 3, so testing the values one at a time is safer.

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