Download Chapter 5.4+Ch1.3: Random Variable and Its Probability Distribution

Ch5.4 + Ch1.3 Random Variable and Its Probability Distribution:
Part II: Continuous Random Variable
----------------------------------------------------------------------------------------------------------Topics: Probability Distribution of a continuous random variable (§5.4, §1.3)
Mean and Variance of a continuous random variable (§5.4, §2.1, §2.2)
---------------------------------------------------------------------------------------------------------- Probability distribution of continuous random Variable (r.v.)
 A continuous random variable is a random variable that can take every
possible value in a certain interval.
Ex. Let x = tomorrow’s temperature in Raleigh. Then technically, x can be
anything in (, ) . Of course, x does not take any value with equal
probability. We need a function to describe its distribution.
 Probability density function for x: f ( x ) satisfies:
(1) f ( x)  0, for any x
(2)



f ( x)dx  1 (that is, P[  x  ]  1.
0.04
0.02
0.00
density function
0.06
0.08
Graphically (the area represents P[55  x  62] )
45
50
55
60
65
70
75
temperature (x)
1
Ex. Suppose the life time x (in months) of a particular type of light bulbs
produced by GE has a probability density function
1  4x
f ( x)  e , x  0
4
(a) Check f(x) is a density function and find (b) a randomly purchased light
bulb will work less than 1 month; (c) a randomly purchased light bulb will work
after being used for 5 months.
(a) Obviously f ( x )  0 .



f ( x)dx  

0
x

1  4x
e dx  e 4
4
0

 1.
(b)
x

1  4x
4
P[ x  1]   e dx  e
0 4
1
0
1
 1  e1/ 4  0.22
(c)
P[ x  5]  

5
x

1  4x
e dx  e 4
4
5

 e5/ 4  0.29
Ex. Suppose a continuous random variable x has a density in the following
form
f ( x)  kx 2 , 0  x  1
Find the constant k and calculate P[x<0.5] and P[0.1  x  0.6] .

1
0
1
k
k
f ( x)dx   kx 2 dx  x 3 10   1, so k=3.
0
3
3
 Mean and variance of a continuous random variable x with density f(x)
1. Mean of x:
 x    xf ( x)dx
Ex. The GE light bulb example:
x  


xf ( x)dx  

0
x

1 x
x e 4 dx   xe 4
4

0


x
4
  e dx  4 (months)
0
2
Ex. For the second example,
1
1
0
0
 x   x f ( x)dx  3 x3dx 
3 4
x
4
1
0

3
4
2. Variance of x (with probability density function f(x))




The variances  x2   ( x  x )2 f ( x)dx   x2 f ( x)dx  x2
The standard deviation  x   x2
Ex. The GE light bulb example:

   x f ( x)dx    
2
x

2
2
x

0
1  4x
x  e dx  42  32  16  16
4
2
 x   x2  16  4 .
3