For the random variable dependent on one another requires the calculation of conditional probabilities which we already discussed, now we will discuss some more parameters for such random variables or experiments like conditional expectation and conditional variance for different types of random variables.
Conditional Expectation
The definition of conditional probability mass function of discrete random variable X given Y is
here pY(y)>0 , so the conditional expectation for the discrete random variable X given Y when pY (y)>0 is
in the above expectation probability is the conditional probability.
In similar way if X and Y are continuous then the conditional probability density function of the random variable X given Y is
where f(x,y) is joint probability density function and for all yfY(y)>0 , so the conditional expectation for the random variable X given y will be
for all yfY(y)>0.
As we know that all the properties of probability are applicable to conditional probability same is the case for the conditional expectation, all the properties of mathematical expectation are satisfied by conditional expectation, for example conditional expectation of function of random variable will be
and the sum of random variables in conditional expectation will be
Conditional Expectation for the sum of binomial random variables
To find conditional expectation of the sum of binomial random variables X and Y with parameters n and p which are independent, we know that X+Y will be also binomial random variable with the parameters 2n and p, so for random variable X given X+Y=m the conditional expectation will be obtained by calculating the probability
since we know that
thus the conditional expectation of X given X+Y=m is
Example:
Find the conditional expectation
if the joint probability density function of continuous random variables X and Y is given as
solution:
To calculate the conditional expectation we require conditional probability density function, so
since for the continuous random variable the conditional expectation is
hence for the given density function the conditional expectation would be
Expectation by conditioning||Expectation by conditional expectation
We can calculate the mathematical expectation with the help of conditional expectation of X given Y as
for the discrete random variables this will be
which can be obtained as
and for the continuous random we can similarly show
Example:
A person is trapped in his building underground as the entrance is blocked due to some heavy load fortunately there are three pipelines from which he can come out the first pipe take him safely out after 3 hours, the second after 5 hours and the third pipeline after 7 hours, If any of these pipeline chosen equally likely by him, then what would be the expected time he will come outside safely.
Solution:
Let X be the random variable that denote the time in hours until the person came out safely and Y denote the pipe he chooses initially, so
since
If the person chooses the second pipe , he spends 5 hous in that but he come outside with expected time
so the expectation will be
Expectation of sum of random number of random variables using conditional expectation
Let N be the random number of random variable and sum of random variables is then the expectation
since
as
thus
Correlation of bivariate distribution
If the probability density function of the bivariate random variable X and Y is
where
then the correlation between random variable X and Y for the bivariate distribution with density function is
since correlation is defined as
since the expectation using conditional expectation is
for the normal distribution the conditional distribution X given Y is having mean
now the expectation of XY given Y is
this gives
hence
Variance of geometric distribution
In the geometric distribution let us perform successively independent trials which results in success with probability p , If N represents the time of first success in these succession then the variance of N as by definition will be
Let the random variable Y=1 if the first trial results in success and Y=0 if first trial results in failure, now to find the mathematical expectation here we apply the conditional expectation as
since
if success is in first trial then N=1 and N2=1 if failure occur in first trial , then to get the first success the total number of trials will have the same distribution as 1 i.e the first trial that results in failure with plus the necessary number of additional trials, that is
Thus the expectation will be
since the expectation of geometric distribution is so
hence
and
E
so the variance of geometric distribution will be
Expectation of Minimum of sequence of uniform random variables
The sequence of uniform random variables U1, U2 … .. over the interval (0, 1) and N is defined as
then for the expectation of N, for any x ∈ [0, 1] the value of N
we will set the expectation of N as
to find the expectation we use the definition of conditional expectation on continuous random variable
now conditioning for the first term of the sequence we have
here we get
the remaining number of uniform random variable is same at the point where the first uniform value is y,in starting and then were going to add uniform random variables until their sum surpassed x − y.
so using this value of expectation the value of integral will be
if we differentiate this equation
and
now integrating this gives
hence
the value of k=1 if x=0 , so
m
and m(1) =e, the expected number of uniform random variables over the interval (0, 1) that need to be added until their sum surpasses 1, is equal to e
Probability using conditional Expectation || probabilities using conditioning
We can find the probability also by using conditional expectation like expectation we found with conditional expectation, to get this consider an event and a random variable X as
from the definition of this random variable and expectation clearly
now by conditional expectation in any sense we have
Example:
compute the probability mass function of random variable X , if U is the uniform random variable on the interval (0,1), and consider the conditional distribution of X given U=p as binomial with parameters n and p.
Solution:
For the value of U the probability by conditioning is
we have the result
so we will get
Example:
what is the probability of X < Y, If X and Y are the continuous random variables with probability density functions fX and fY respectively.
Solution:
By using conditional expectation and conditional probability
as
Example:
Calculate the distribution of sum of continuous independent random variables X and Y.
Solution:
To find the distribution of X+Y we have to find the probability of the sum by using the conditioning as follows
Conclusion:
The conditional Expectation for the discrete and continuous random variable with different examples considering some of the types of these random variables discussed using the independent random variable and the joint distribution in different conditions, Also the expectation and probability how to find using conditional expectation is explained with examples, if you require further reading go through below books or for more Article on Probability, please follow our Mathematics pages.
https://en.wikipedia.org/wiki/probability_distribution
A first course in probability by Sheldon Ross
Schaum’s Outlines of Probability and Statistics
An introduction to probability and statistics by ROHATGI and SALEH
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