Chap 1 Continuous random variables
1.1 Piecewise definition of a probability density function
Use probability density function(PDF) to find probabilities
!
𝑃(𝑎 ≤ 𝑥 ≤ 𝑏) = ) 𝑓(𝑥)𝑑𝑥
"
To find the mean, variance, median, percentile, mode from a given PDF
#$
)
𝑓(𝑥)𝑑𝑥 = 1
%$
The expectation and variance of a function of X
For a discrete random variable, X, in which the value xi occurs with probability pi, the
expectation and variance of a function g(X) are given by
𝐸.𝑔(𝑋)1 = 2 𝑔(𝑥& )𝑝&
𝑉𝑎𝑟.𝑔(𝑋)1 = 2{(𝑔(𝑥& ))' 𝑝& } − {𝐸(𝑔(𝑋))}'
The equivalent results for a continuous random variable, X, with PDF f(x) are:
𝐸.𝑔(𝑋)1 =
) 𝑔(𝑥)𝑓(𝑥)𝑑𝑥
())
*")+,./ 1
𝑉𝑎𝑟.𝑔(𝑋)1 =
) (𝑔(𝑥))' 𝑓(𝑥)𝑑𝑥 − {𝐸.𝑔(𝑋)1}'
())
*")+,./ 1
Can think of the function g(X) as a new variable.
1.3 The cumulative distribution function
The function giving 𝑃(𝑋 ≤ 𝑥) is called the cumulative distribution function (CDF), and it
is usually denoted by F(x). The best method of obtaining the cumulative distribution function
is to use indefinite integration.
Change PDF to CDF
1
𝐹(𝑥) = 𝑃(𝑋 ≤ 𝑥) = ) 𝑓(𝑥)𝑑𝑥
%$
Change CDF to PDF:
𝑓(𝑥) =
𝑑
𝐹(𝑥)
𝑑𝑥
To find the median:
𝐹(𝑚) = 0.5, 𝑠𝑜 𝑡ℎ𝑎𝑡, 𝑚 = ?
Notes: pay attention to the range of the variables.
Example:
The continuous random variable X has PDF f(x) given by
𝑓(𝑥) = E
0.2,
0,
𝑓𝑜𝑟 3 ≤ 𝑥 ≤ 8
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
The continuous random variable Y is given by 𝑌 = 5𝑋 ' .
(i) Find H(y) where H(y) is the cumulative distribution function of Y.
(ii) Find h(y).
(iii) Find P(Y<100).
(iv) State the median value of X.
(v) Use H(y) to find the median value of Y, showing your working.
(vi) Show that you can use the equation Y = 5X 2 to find the median value of Y from that of
X.
(vii) Write down the value of E(X) and find E(Y ). Can you use the equation 𝑌 = 5𝑋 ' to
find the mean value of Y from that of X?
Key points:
#$
1. ∫%$ 𝑓(𝑥)𝑑𝑥 = 1
2. 𝑓(𝑥) ≥ 0,
𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥
2
3. 𝑃(𝑐 ≤ 𝑥 ≤ 𝑑) = ∫3 𝑓(𝑥)𝑑𝑥
4. E(𝑥) = ∫ 𝑥𝑓(𝑥)𝑑𝑥
5. Var(𝑥) = ∫ 𝑥 ' 𝑓(𝑥)𝑑𝑥 − [𝐸(𝑋)]'
4
4
6. the median m of X is the value for which: ∫%$ 𝑓(𝑥)𝑑𝑥 and ∫%$ 𝑓(𝑥)𝑑𝑥 = 0.5
7. the mode of X is the value for which f(x) has its greatest magnitude
8. A probability density function may be defined piecewise.
9. If g[X] is a function of X then the expectation and variance of X are:
𝐸.𝑔(𝑋)1 = ) 𝑓(𝑥)𝑔(𝑥)𝑑𝑥
𝑉𝑎𝑟.𝑔(𝑋)1 = )(𝑔(𝑥))' 𝑓(𝑥)𝑑𝑥 − (𝐸.𝑔(𝑋)1)'
10. If f(x) is the probability density function of X then the cumulative distribution function
(CDF) of X is F(x) where:
1
(i) 𝐹(𝑥) = ∫" 𝑓(𝑢)𝑑𝑢 where the constant a is the lower limit of X
2
(ii) 𝑓(𝑥) = 21 𝐹(𝑥)
(iii) for the median, m, F(m)=0.5
11. Given that f(x) is the probability density function of X, you can find that of a related
variable.
To do this you need first to find the cumulative distribution function of X and use this to find
that of Y. You can then differentiate the cumulative distribution function of Y to find f(y), the
probability density function of Y.