5. Binomial Distribution & Discrete Random Variables
Discrete Random Variables
3:26 minutesProblem 4.1.43
Textbook Question
Independent and Dependent Random Variables Two random variables x and y are independent when the value of x does not affect the value of y. When the variables are not independent, they are dependent. A new random variable can be formed by finding the sum or difference of random variables. If a random variable x has mean and a random variable y has mean , then the means of the sum and difference of the variables are given by . If random variables are independent, then the variance and standard deviation of the sum or difference of the random variables can be found. So, if a random variable x has variance and a random variable y has variance , then the variances of the sum and difference of the variables are given by In Exercises 43 and 44, the distribution of SAT mathematics scores for college-bound male seniors in 2020 has a mean of 531 and a standard deviation of 121. The distribution of SAT mathematics scores for college-bound female seniors in 2020 has a mean of 516 and a standard deviation of 112. One male and one female are randomly selected. Assume their scores are independent. (Adapted from College Board)
Find the mean and standard deviation of the sum of their scores.
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Step 1: Understand the problem. We are tasked with finding the mean and standard deviation of the sum of the SAT mathematics scores for one randomly selected male and one randomly selected female. The problem states that the scores are independent, which is crucial for applying the formulas for the mean and variance of the sum of independent random variables.
Step 2: Recall the formula for the mean of the sum of two independent random variables. If X and Y are two independent random variables with means μ_X and μ_Y, then the mean of their sum (X + Y) is given by: = . Use the given means for males (531) and females (516) to calculate the mean of the sum.
Step 3: Recall the formula for the variance of the sum of two independent random variables. If X and Y are independent random variables with variances σ²_X and σ²_Y, then the variance of their sum (X + Y) is given by: = . Use the given standard deviations for males (121) and females (112) to calculate the variance of the sum. Remember that variance is the square of the standard deviation.
Step 4: Calculate the standard deviation of the sum. The standard deviation is the square root of the variance. Use the variance calculated in Step 3 to find the standard deviation of the sum: = .
Step 5: Summarize the results. The mean of the sum is the value calculated in Step 2, and the standard deviation of the sum is the value calculated in Step 4. These are the final results for the mean and standard deviation of the sum of the SAT mathematics scores for one male and one female.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Independent Random Variables
Independent random variables are those whose outcomes do not influence each other. This means that knowing the value of one variable provides no information about the value of the other. For example, if X represents the score of a male student and Y represents the score of a female student, their independence implies that the male's score does not affect the female's score.
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Mean of Random Variables
The mean of a random variable is a measure of its central tendency, representing the average value expected from the variable. When combining independent random variables, the mean of their sum is simply the sum of their individual means. For instance, if the mean score of males is 531 and that of females is 516, the mean of their combined scores would be 531 + 516.
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Variance and Standard Deviation of Random Variables
Variance measures the spread of a set of values around the mean, while standard deviation is the square root of variance, providing a measure of dispersion in the same units as the data. For independent random variables, the variance of their sum is the sum of their variances. Thus, if the variances of the male and female scores are known, the variance of their combined scores can be calculated by adding these variances together.
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