Alg1231W12T3 - Continuous distributions

Sunday, 21 October 2018

12:06 AM

Machine generated alternative text:
For x < 0 we have fn(x) 
This proves that fn (x) 2 0 
Integrating the density over R gives 
For any natural number n 
• the function 
X for x > 0 
cne¯Xdæ 
o 
otherwise 
is a probability density function We prove this by showing that fn(x) is not negative and it integrates to one 
O . And for x 2 0 we have [select all that apply] 
O for all x. 
fn (x)dc 
DO 1 
where a 
O 
It can be shown (using integration by parts) that 
If we use this result; then we can conclude that 
as required 
fn (x)dx

 

Machine generated alternative text:
Consider the Laplace distribution 
Since the exponential function is 
e ¯tdx 
1 
f@) 
2 
for 
—DO < < DO 
positive tor all real inputs 
O we can easily see that 2 
f(x) dc 
O for all x e R. 
Integrating this density over R we get 
Because the density is an even function 
1 
e 
¯lxldx. 
O 
, we can conclude that 
where a 
as required 
O 
We can caculate this as an improper integral, showing 
e¯Xdx = lim 
2 
—exp(-R) + 1

 

Machine generated alternative text:
Let's calculate the mean and variance of the Gamma distributiom the probability density function of the Gamma distribution is defined to be 
X n —x for x > 0 
o 
In general, the mean of a continuous random variable is defined to be 
So, in the case of the gamma distribution this is 
otherwise. 
xfn@) dc. 
oo 1 
n+1e¯Xdx. 
Using a well know result (see note below) we can easily calculate this to be 
The variance can be calculate by 
Var(X) 
however it is much easier to calculate it using the alternative form 
Var(X) = E(X2) - E(X)2. 
The expectation of the X2 is given by 
n+2e¯Xdx. 
E(X2) — 
Use the same result as before (see note below) we can easily calculate this to be 
From this we calculate that 
Note: Recall that important formula 
for m 
E(X2) — 
Var(X) = E(X2) - 
n+l 
e

 

Machine generated alternative text:
For a parameter a > 0 the probability density function of the Pareto distribution is defined to be 
fa(.c) — 
for x > 1 
otherwise. 
O 
An interesting fact about the Pareto distribution is that the mean and variance is not defined for some choices of the parameter a. 
The mean of a continuous random variable with this distribution will be 
xfa@) dc 
Using the test from p-integrals from calculus, we know that this improper integral will only converge to a finite number when a > 
Note: you can type a in Maple notations by typing alpha. 
As before we'll calculate the variance using the formula 
Var(X) = E(X2) - E(X)2. 
The expectation of the X2 is given by 
E(X2) - 
Like before we know this will only converge to a finite number when a — 1 > 1 
1 
O 
a—I 
, or in other words, when a > 
2 
O 
O 
In this case we can find 
Calculating we get 
E(X2) ¯ 
alphai(alpfia-2) 
From this we can calculate that 
Var(X) 
(a—2)(a 
-1)2

 

Machine generated alternative text:
-2 
0.8 
0.6 
0.2 
S Z 1) 0.341 
2 
z 
3 
The GeoGebra app above shows the probabilty density function for the standard normal distribution N(O, 1) _ The probability density function for this distribution is given by the expression 
1 
2K 
Answer the following questions about a random variable Z which has the standard normal distribution (i.e. Z 
i) P(—l < Z < 1) is about 68 
ii) P(—2 < Z < 2) is about 95 
iii) P(—1.5 < Z < 2.5) is about 93 
iv)P(0.5 < Z < 1.5) is about 24 
O 
O 
percent Note that this corresponds to the points within 1 
percent Note that this corresponds to the points within 2 
O percent 
O percent 
O standard deviation of the mean. 
O standard deviations of the mean

 

 

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