Answer :
the mean and standard deviation of the random variable X, which represents the income from insuring four 21-year-old men, is 2000 and 27.22, respective
Mean: $2,000
Standard Deviation: $741.42
The mean of the random variable X is calculated by taking the sum of the products of the probability of each outcome and the corresponding value of X. The probabilities of each outcome, in this case, are 0, 0.38, 0.59, and 0.03, respectively. The corresponding values of X are $1000, 2000, 3000, and 4000$. Thus, the mean of X is calculated as:
Mean =[tex]$\sum_{i=1}^{N} P(X = x_i)*x_i[/tex]
= 0*1000 + 0.38*2000 + 0.59*3000 + 0.03*4000
= 2000
The standard deviation of the random variable X is calculated by taking the square root of the sum of the products of the probability of each outcome and the square of the difference between the corresponding value of X and the mean. The probabilities of each outcome, in this case, are 0, 0.38, 0.59, and 0.03, respectively. The corresponding values of X are $1000, 2000, 3000, and 4000$. The mean of X is 2000. Thus, the standard deviation of X is calculated as:
Standard Deviation = [tex]$\sqrt{\sum_{i=1}^{N} P(X = x_i)(x_i - \mu)^2}[/tex]
= [tex]\sqrt{(0(1000-2000)^2[/tex] + [tex]0.38*(2000-2000)^2[/tex] + [tex]0.59*(3000-2000)^2 + 0.03*(4000-2000)^2)}[/tex]
= [tex]\sqrt{741.42} = 27.22$[/tex]
Therefore, the mean and standard deviation of the random variable X, which represents the income from insuring four 21-year-old men, is 2000 and 27.22, respective
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