(12)continuous distributions
Post on 14-Apr-2018
214 Views
Preview:
TRANSCRIPT
-
7/30/2019 (12)Continuous Distributions
1/7
Applied Statistics and Computing Lab
CONTINUOUS DISTRIBUTIONS
Applied Statistics and Computing Lab
Indian School of Business
-
7/30/2019 (12)Continuous Distributions
2/7
Applied Statistics and Computing Lab
Probability Distribution of a Continuous
Random Variable
Weights of new born babies is an example of continuous randomvariable.
What is the probability of the weight of a new born baby beingexactly 3.54689 kgs, given that the weights of all new born babies
lie in the range 2.5-4 kgs? Answer is 0- probability of one event occurring out of an
uncountably infinite number of possibilities
Concept of pmf not valid- pmf is the probability of X assuming aparticular value x
But what if you are asked the probability of weight lying in theinterval 3.5-4, which includes the number 3.54689?
Concept of probability density function- based on finding theprobability of a continuous random variable lying in a particularinterval
2
-
7/30/2019 (12)Continuous Distributions
3/7
Applied Statistics and Computing Lab
Probability Density Function
3
For a continuous random variable, if there exists a function f(x), such that, for
ab ,
( This gives the probability that X lies in the interval
[a,b])
A function of this type satisfying the conditionsa) f(x)0, for all x
b)
Is called the probability density function of x
( ) 1f x dx
( ) ( )b
a
f x dx P a x b
-
7/30/2019 (12)Continuous Distributions
4/7
Applied Statistics and Computing Lab
Revisiting the new born baby example
Now,
We are interested in finding P(axa)=
P(X=a)=
Hence the probability of a continuous variable
taking a single value is 0
Thus, the probability of the weight of a newborn baby being exactly 3.54689 kgs is 0
4
( ) 0
a
a
f x dx
( ) ( )b
a
f x dx P a x b
-
7/30/2019 (12)Continuous Distributions
5/7
Applied Statistics and Computing Lab
Cumulative Distribution function, Expectation,
Variance of a Random Variable: The Continuous
counterpart All the concepts and definitions are the same as those for discrete random variable In all the definitions, the summation sign is simply replaced by its continuous
counterpart- the integral sign
Cumulative Distribution Function:
F(x)= P(X x)=
Expectation: E(X)=
Variance: v(X)=
5
x
(- )
( )f x dx
x
(- )
( )xf x dx
x
2
(- )
( ( )) ( )x E X f x dx
-
7/30/2019 (12)Continuous Distributions
6/7
Applied Statistics and Computing Lab
Applications A clock can stop at any time during the day. Let X be the time (hours plus
fractions of hours) at which the clock stops. The pdf for X isf (x) =(1/24), 0x24
=0, otherwise
What is the probability that the clock will stop between 2:00pm and 2:45pm?
P(14X14.75)=
Let the pdf of a continuous random variable x be
f(x) = (1/2)-(x/8), 0x4
=0, otherwise
a) What is P(2x+3>5) b) What is E(X)?(Hint: a) P(2x+3>5) is equivalent to P(x>1).
b) E(X)= )
6
14.75
14
1 1
24 32dx
4
0
1( )
2 8
xx dx
-
7/30/2019 (12)Continuous Distributions
7/7Applied Statistics and Computing Lab
Thank you
top related