Probability

If events A and B are independent, we have the property:

Given:

Let . Then,

Thus,

Taking the square root on both sides:

Therefore, .

We need to determine the variance of a normal variate given its moment-generating function (MGF) .

1. Understanding the Moment-Generating Function:
For a normal random variable , the MGF is:


where is the mean and is the variance. We are given , which we can compare to the standard form.

2. Comparing the Given MGF:
The given MGF is:


Matching this with the standard form , we identify the exponents:
- The term with : , so .
- The term with : , so .

3. Solving for Variance:
From , multiply both sides by 2:


The variance of is .

Final Answer:
The variance of the normal variate is .

Step 1: Understand the Problem
We need to find the probability of drawing two red balls in succession without replacement. The total number of balls is . Since the draws are without replacement, the probability of the second draw depends on the first.

Step 2: Calculate the Probability
The probability that both balls are red is the product of: - The probability that the first ball is red. - The probability that the second ball is red, given the first is red.
- First draw: There are 5 red balls out of 12 total balls.
- Second draw: After drawing one red ball, 4 red balls and 11 total balls remain. The joint probability is: Simplify the fraction:

Step 3: Final Answer
The probability that both balls are red is: