Final answer:
The support of the random variable X in the mixture distribution is the union of the supports of the individual distributions f₁₁ (x) and f₁₂ (x), consisting of all values where these functions are positive.
Step-by-step explanation:
The support of a random variable X in a mixture distribution is determined by the range of values over which the probability mass function (pmf) is positive. In the given mixture distribution f₁ (x)=.7f₁₁ (x)+.3f₁₂ (x), the support of X is the union of the supports of the individual distributions f₁₁ (x) and f₁₂ (x). The support of X will include all the values of x for which either f₁₁ (x) or f₁₂ (x) is positive, considering that a mixture probability of 0.7 and 0.3 has been assigned to them respectively. To determine the exact support, one would examine the supports of f₁₁ (x) and f₁₂ (x) to see over which range of x values these functions are defined and positive.