`=>` 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 `f_4` 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:
`color{blue} "Definition "`
`=>` 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_i >0 sum_(i=1)^n p_i = 1 , =1,2, ...., n`
`=>` The real numbers `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 `x_i` i.e., `P(X = x_i) = p_i`
`color{blue}{"Key Point"} Note` :- If `x_i` 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 `x_i` 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 `f_4` 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:
`color{blue} "Definition "`
`=>` 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_i >0 sum_(i=1)^n p_i = 1 , =1,2, ...., n`
`=>` The real numbers `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 `x_i` i.e., `P(X = x_i) = p_i`
`color{blue}{"Key Point"} Note` :- If `x_i` 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 `x_i` 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.