Let us look at the experiment of selecting one family out of ten families `f_1, f_2 ,..., f_(10)` in
such a manner that each family is equally likely to be selected. Let the families `f_1, f_2,
... , f_(10)` have `3, 4, 3, 2, 5, 4, 3, 6, 4, 5` members, respectively.
Let us select a family and note down the number of members in the family denoting
X. Clearly, X is a random variable defined as below :
`X(f_1) = 3, X(f_2) = 4, X (f_3) = 3, X(f_4) = 2, X (f_5) = 5,`
`X(f_6) = 4, X(f_7) = 3, X (f_8) = 6, X(f_9) = 4, X(f_(10)) = 5`
Thus, X can take any value 2,3,4,5 or 6 depending upon which family is selected.
Now, X will take the value 2 when the family f4 is selected. X can take the value
3 when any one of the families `f_1, f_3, f_7` is selected.
Similarly, `X = 4,` when family `f_2, f_6` or `f_9` is selected,
`X = 5,` when family `f_5` or `f_(10)` is selected
and `X = 6,` when family `f_8` is selected.
Since we had assumed that each family is equally likely to be selected, the probability
that family `f_4` is selected is `1/10`
.
Thus, the probability that X can take the value 2 is`1/10` . We write P(X = 2) =`1/10`
Also, the probability that any one of the families `f_1, f_3` or `f_7` is selected is
`P({f_1, f_3, f_7}) =3/10`
Thus, the probability that X can take the value `3 = 3/10`
We write `P(X = 3) =3/10`
Similarly, we obtain
`P(X = 4) = P({f_2, f_6, f_9}) =3/10`
`P(X = 5) = P({f_5 , f_10}) = 2/10`
and `P(X =6) = P({f_8}) = 1/10`
Such a description giving the values of the random variable along with the
corresponding probabilities is called the probability distribution of the random
variable X.
In general, the probability distribution of a random variable X is defined as follows:;
`text(Definition 5)` The probability distribution of a random variable X is the system of numbers
`tt((X , : , x_1 , x_2 , ....,x_n),(P(X),: ,p_1,p_2,....,p_n))`
where `p(x_i) = sum_(i=1)^n`
`p_i = 1 , i=1,2,......,n`
The real number `x_1 , x_2 , ............., x_n` are the possible values of the random variable X and
`p_i(i= 1.2,......,n)` is the probability of the random variable X taking the value xi i.e.,
`P(X = x_i) = p_i`
`text(Note)` If xi is one of the possible values of a random variable X, the statement
`X = x_i` is true only at some point (s) of the sample space. Hence, the probability that
X takes value xi is always nonzero, i.e. `P(X = x_i) ≠ 0` .
Also for all possible values of the random variable X, all elements of the sample
space are covered. Hence, the sum of all the probabilities in a probability distribution
must be one.
Let us look at the experiment of selecting one family out of ten families `f_1, f_2 ,..., f_(10)` in
such a manner that each family is equally likely to be selected. Let the families `f_1, f_2,
... , f_(10)` have `3, 4, 3, 2, 5, 4, 3, 6, 4, 5` members, respectively.
Let us select a family and note down the number of members in the family denoting
X. Clearly, X is a random variable defined as below :
`X(f_1) = 3, X(f_2) = 4, X (f_3) = 3, X(f_4) = 2, X (f_5) = 5,`
`X(f_6) = 4, X(f_7) = 3, X (f_8) = 6, X(f_9) = 4, X(f_(10)) = 5`
Thus, X can take any value 2,3,4,5 or 6 depending upon which family is selected.
Now, X will take the value 2 when the family f4 is selected. X can take the value
3 when any one of the families `f_1, f_3, f_7` is selected.
Similarly, `X = 4,` when family `f_2, f_6` or `f_9` is selected,
`X = 5,` when family `f_5` or `f_(10)` is selected
and `X = 6,` when family `f_8` is selected.
Since we had assumed that each family is equally likely to be selected, the probability
that family `f_4` is selected is `1/10`
.
Thus, the probability that X can take the value 2 is`1/10` . We write P(X = 2) =`1/10`
Also, the probability that any one of the families `f_1, f_3` or `f_7` is selected is
`P({f_1, f_3, f_7}) =3/10`
Thus, the probability that X can take the value `3 = 3/10`
We write `P(X = 3) =3/10`
Similarly, we obtain
`P(X = 4) = P({f_2, f_6, f_9}) =3/10`
`P(X = 5) = P({f_5 , f_10}) = 2/10`
and `P(X =6) = P({f_8}) = 1/10`
Such a description giving the values of the random variable along with the
corresponding probabilities is called the probability distribution of the random
variable X.
In general, the probability distribution of a random variable X is defined as follows:;
`text(Definition 5)` The probability distribution of a random variable X is the system of numbers
`tt((X , : , x_1 , x_2 , ....,x_n),(P(X),: ,p_1,p_2,....,p_n))`
where `p(x_i) = sum_(i=1)^n`
`p_i = 1 , i=1,2,......,n`
The real number `x_1 , x_2 , ............., x_n` are the possible values of the random variable X and
`p_i(i= 1.2,......,n)` is the probability of the random variable X taking the value xi i.e.,
`P(X = x_i) = p_i`
`text(Note)` If xi is one of the possible values of a random variable X, the statement
`X = x_i` is true only at some point (s) of the sample space. Hence, the probability that
X takes value xi is always nonzero, i.e. `P(X = x_i) ≠ 0` .
Also for all possible values of the random variable X, all elements of the sample
space are covered. Hence, the sum of all the probabilities in a probability distribution
must be one.
