International A Level (IAL)MathsEdexcelStatistics 2FlashcardsStatistical DistributionsCumulative Distribution Function

Cumulative Distribution Function (Edexcel International A Level (IAL) Maths: Statistics 2): Flashcards

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Define the cumulative distribution function of a continuous random variable.

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Define the cumulative distribution function of a continuous random variable.
It is $F (x_{0}) = P (X \leq x_{0})$ , the probability that $X$ takes a value less than or equal to $x_{0}$ .
On the graph of the probability density function this is the area under the curve up to the vertical line at $x = x_{0}$ .
What is the difference between $F (x)$ and $f (x)$ ?
The capital $F$ is always the cumulative distribution function and the lower-case $f$ is always the probability density function, and the two are never used the other way round.
$F (x)$ is an accumulated probability, so it always lies between 0 and 1, whereas $f (x)$ is a density and is not a probability at all.
That is why $f (x)$ is allowed to be greater than 1 while $F (x)$ never is.
True or False?
For a continuous random variable $P (X = k) = 0$ for every $k$ , so $F (k)$ must be 0 as well.
False.
$F (k)$ is $P (X \leq k)$ , the whole accumulated probability up to $k$ , and not the probability of the single value $k$ .
So a variable can perfectly well have $P (X = 2) = 0$ and $F (2) = 0.75$ at the same time, and this slips past people far more easily when working with $F$ than with $f$ .
One of only two formulae in this module that the formula booklet does not give you links $f$ and $F$ . Complete both directions:
$F (x) = \int_{- \infty}^{x}______d t$
$f (x) = \frac{d}{d x}______$
The completed formulae are:
$F (x) = \int_{- \infty}^{x} f (t) d t$
$f (x) = \frac{d}{d x} F (x)$
Integrating takes you from the density to the accumulated probability, and differentiating brings you straight back the other way.
The dummy variable $t$ in the first is there only because $x$ is already being used as the upper limit.
True or False?
The graph of a cumulative distribution function is always continuous, even when $F$ is defined piecewise.
True.
Because $P (X = k) = 0$ for a continuous variable, no probability is ever added in a single jump, so $F$ cannot step upwards anywhere.
It climbs without breaks from 0 on the left to 1 on the right, so where two pieces meet their values must agree, which is the quickest check on a piecewise $F$ .
Once you have $F (x)$ , how do you find $P (a \leq X \leq b)$ ?
Subtract one value of $F$ from the other:
$P (a \leq X \leq b) = F (b) - F (a)$
Everything accumulated up to $b$ , less everything accumulated up to $a$ , leaves exactly the probability in between.
Once $F$ is known no integration is needed at all, which is what makes finding $F$ first worth the effort when several probabilities are wanted.
You are building $F (x)$ from a piecewise $f (x)$ . Why is integrating each piece between its own limits not enough?
Because $F$ accumulates, so each piece has to start from the total already reached at the end of the piece before it.
For a second piece beginning at $x = a$ that gives:
$F (x) = F (a) + \int_{a}^{x} f (t) d t$
Leaving out the $F (a)$ is the usual error, and it shows up as a function that fails to reach 1 at the top of the range.
Part of a cumulative distribution function is constant over an interval. What does that tell you about the variable there?
That $X$ never takes a value in that interval, because no probability at all is being accumulated across it.
The probability density is zero right through the interval, so the graph of $f (x)$ has a gap exactly where the graph of $F (x)$ is flat.
A flat stretch of $F$ is the c.d.f.'s way of showing a hole in the range of the variable.
How do you find the median and the lower quartile of $X$ from its cumulative distribution function?
Solve $F (m) = 0.5$ for the median and $F (Q_{1}) = 0.25$ for the lower quartile, and $F (p) = \frac{n}{100}$ for the $n$ th percentile.
Because $F$ gives the probability below a value directly, these are ordinary equations to solve rather than integrals to evaluate.
For a piecewise $F$ , work out its value at the end of each piece first, so that you know which piece the answer lies in before solving anything.
The continuous uniform distribution on $a \leq x \leq b$ has $f (x) = \frac{1}{b - a}$ . Find its cumulative distribution function.
For $a \leq x \leq b$ , $F (x)$ is the area of the rectangle from $a$ up to $x$ , which is $(x - a) \times \frac{1}{b - a}$ :
$F (x) = \frac{x - a}{b - a}$
A full answer also states that $F (x) = 0$ for $x < a$ and $F (x) = 1$ for $x > b$ .
A rectangular density therefore gives a c.d.f. that climbs in a straight line, which is the simplest cumulative distribution function there is.

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