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Probability Density Function (Edexcel International A Level Maths: Statistics 2)
Revision Note
Calculating Probabilities using PDF
What is a probability density function (p.d.f.)?
- For a continuous random variable, it is often possible to model probabilities using a function
- This function is called a probability density function (p.d.f.)
- For the continuous random variable, X , it would usually be denoted as a function of x (such as f(x) or g(x) ) and is usually given piecewise
e.g.
- f(x) should be defined for all values of
- The distribution (or density) of probabilities can be illustrated by the graph of f(x)
- The graph does not need to start and end on the x-axis
- The graph does not have to be continuous
- For f(x) to represent a p.d.f. the following conditions must apply
- for all values of x
- This is the equivalent to for a discrete random variable
- The area under the graph must total 1
- for all values of x
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- This is equivalent to for a discrete random variable
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How do I find probabilities using a probability density function (p.d.f.)?
- The probability that the continuous random variable X lies in the interval
, where X has the probability density function f(x) , is given by
-
- As with the normal distribution
- for any continuous random variable, for all values of n
- One way to think of this is that in the integral above
- As with the normal distribution
- Piecewise Function are often used as the p.d.f. is not often a single function of x
- Finding a probability may involve splitting the area across more than one piece of the function
- This will depend on the limits a and b in that is being found
How do I solve problems using the PDF?
- Some questions may ask for justification of the use of a given function for a probability density function
- In such cases check that the function meets the two conditions for all values of and the total area under the graph is 1
- If asked to find a probability
- STEP 1
As the probability density function, f(x), is usually given piecewise make sure you are clear about the values of x for which each part applies
- STEP 1
e.g.
- STEP 2
If simple to do so, sketching the graph of y= f(x) may help to find the probability
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-
- Look for basic shapes such as triangles or rectangles; finding areas of these is easy and avoids integration
- Look for symmetry in the graph that may make the problem easier
-
- STEP 3
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- Identify the range of X, particularly noting if it is split across different parts of the p.d.f.
- Find the required area (probability), either by basic shapes or integrate f(x) and evaluate it between the two limits, splitting if necessary
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- Trickier problems may involve finding a limit of the integral given its value
- i.e. one of the values in the range of X, given the probability
e.g. Find the value of given
- i.e. one of the values in the range of X, given the probability
Worked example
The continuous random variable, , has probability density function
Examiner Tip
- If the graph is easy to draw, then a sketch of f(x) is helpful
- Some p.d.f. graphs have symmetry, common shapes such as triangles or rectangles so areas are easier to find, avoiding the need for integration
- Always keep an eye for probabilities that are split across different parts of a piecewise function
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Median and Mode of a CRV
What is meant by the median of a continuous random variable?
- The median, m, of a continuous random variable, X , with probability density function f(x) is defined as the value of the continuous random variable X, such that
- Since this can also be written as
- If the p.d.f. is symmetrical (i.e. the graph of y = f(x) is symmetrical) then the median will be halfway between the lower and upper limits of x
- In such cases the graph of y=f(x) has axis of symmetry in the line x = m
How do I find the median of a continuous random variable?
- By solving one of the equations to find m
and
- If the graph of is symmetrical, symmetry may be used to deduce the median
- For piecewise functions, you will need to determine which part of the function the median lies within to determine which equation to use
- If there are more than two (non-zero) parts to a function then the integration may need splitting
- You can also use the cumulative distribution function to find the median
How do I find quartiles (or percentiles) of a continuous random variable?
- In a similar way to finding the median
- The lower quartile will be the value L such that P(X ≤ L) = 0.25 or
P(X ≥ L) = 0.75 - The upper quartile will be the value U such that P(X ≤ U) = 0.75 or
P(X ≥ U) = 0.25
- The lower quartile will be the value L such that P(X ≤ L) = 0.25 or
- Percentiles can be find in the same way
- The 15th percentile will be the value k such that P(X ≤ k) = 0.15 or
P(X ≥ k) = 0.85
- The 15th percentile will be the value k such that P(X ≤ k) = 0.15 or
- In all cases start by determining which part(s) of the function are involved
What is meant by the mode of a continuous random variable?
- The mode of a continuous random variable, X , with probability density function f(x) is the value of x that produces the greatest value of f(x) .
How do I find the mode of a PDF?
- This will depend on the type of function f(x); the easiest way to find the mode is by considering the shape of the graph of y= f(x)
- If the graph is a curve with a (local) maximum point, the mode can be found by differentiating and solving the equation f'(x) = 0
- If there is more than one solution to f'(x) = 0 , further work may be needed to deduce which answer is the mode
- Look for valid values of from the definition of the p.d.f.
- Use the second derivative (f'' (x) ) to deduce the nature of each stationary point
- You may need to check the values of f(x) at the endpoints too
- If there is more than one solution to f'(x) = 0 , further work may be needed to deduce which answer is the mode
Worked example
The continuous random variable has probability density function defined as
Examiner Tip
- Avoid spending too long sketching the graph of y = f(x), only do this if the graph is straightforward as finding the median and mode by other means can be just as quick
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