Final answer:
To calculate the pmf of X, we consider two scenarios: drawing with replacement (binomial distribution) when a red ball is drawn and without replacement when a blue ball is drawn from the box. The mean and variance require using formulas specific to each distribution involved (binomial for red, hypergeometric for blue).
Step-by-step explanation:
The student is working with a probability question involving a two-stage random experiment with different conditions for ball selection depending on the outcome of the first draw. To find the probability mass function (pmf) of X, we need to calculate the probability of drawing a certain number of white balls given the initial condition (drawing a red or blue ball from the box).
When a red ball is drawn from the box, the balls from the urn are selected with replacement, meaning the probability of drawing a white ball remains the same for each of the 10 draws, which is 10/25 or 2/5. The number of white balls drawn in this case follows a binomial distribution with parameters n=10 and p=2/5.
When a blue ball is drawn from the box, the balls from the urn are selected without replacement. The probability changes with each draw since the composition of the urn changes, making the calculations more complex. It involves finding the probability of all possible combinations of white and black balls drawn without replacement.
The mean and variance of X can be found using the formulas for the expectation and variance of the binomial distribution in the case of drawing a red ball, and by applying the definitions of mean and variance for hypergeometric distribution in the case of drawing a blue ball.
To compute the exact pmf, mean, and variance, you would use combinations and the appropriate probability formulas for each scenario (with and without replacement).