Question

lesson 6.4 homework

1. the means and standard deviations of independent random variables, s and t, are shown in the table.

random variable	mean	standard deviation
t	53	6.5

a. let random variable u be defined as u = t + 15. find the mean and standard deviation of u.

b. let random variable v be defined as v = s + t. find the mean and standard deviation of v.

c. let random variable w be defined as w = t - s. find the mean and standard deviation of w.

d. let random variable x be defined as x = 2(s). find the mean and standard deviation of x.

Explanation:

Step1: Mean of \(U\)

For a random variable \(U = T + c\) (where \(c\) is a constant), the mean \(\mu_U=\mu_T + c\). Given \(\mu_T = 53\) and \(c = 15\), then \(\mu_U=53 + 15=68\).

Step2: Standard deviation of \(U\)

For a random variable \(U = T + c\), the standard deviation \(\sigma_U=\sigma_T\). Since \(\sigma_T = 6.5\), so \(\sigma_U = 6.5\).

Step3: Mean of \(V\)

For \(V=S + T\), the mean \(\mu_V=\mu_S+\mu_T\). Given \(\mu_S = 34.8\) and \(\mu_T = 53\), then \(\mu_V=34.8+53 = 87.8\).

Step4: Standard deviation of \(V\)

Since \(S\) and \(T\) are independent, for \(V = S+T\), the variance \(\text{Var}(V)=\text{Var}(S)+\text{Var}(T)\). We know that \(\text{Var}(S)=\sigma_S^{2}=(7.9)^{2}=62.41\) and \(\text{Var}(T)=\sigma_T^{2}=(6.5)^{2}=42.25\). Then \(\text{Var}(V)=62.41 + 42.25=104.66\), and \(\sigma_V=\sqrt{104.66}\approx10.23\).

Step5: Mean of \(W\)

For \(W=T - S\), the mean \(\mu_W=\mu_T-\mu_S\). Given \(\mu_S = 34.8\) and \(\mu_T = 53\), then \(\mu_W=53 - 34.8=18.2\).

Step6: Standard deviation of \(W\)

Since \(S\) and \(T\) are independent, for \(W=T - S\), the variance \(\text{Var}(W)=\text{Var}(T)+\text{Var}(S)\) (because \(\text{Var}(aX - bY)=a^{2}\text{Var}(X)+b^{2}\text{Var}(Y)\) with \(a = 1\), \(b = 1\)). \(\text{Var}(S)=62.41\), \(\text{Var}(T)=42.25\), \(\text{Var}(W)=62.41+42.25 = 104.66\), and \(\sigma_W=\sqrt{104.66}\approx10.23\).

Step7: Mean of \(X\)

For \(X = 2S\), the mean \(\mu_X=2\mu_S\). Given \(\mu_S = 34.8\), then \(\mu_X=2\times34.8 = 69.6\).

Step8: Standard deviation of \(X\)

For \(X = 2S\), the standard deviation \(\sigma_X=2\sigma_S\). Since \(\sigma_S = 7.9\), so \(\sigma_X=2\times7.9=15.8\).

Answer:

a. Mean of \(U\): \(68\), Standard deviation of \(U\): \(6.5\)
b. Mean of \(V\): \(87.8\), Standard deviation of \(V\): \(\approx10.23\)
c. Mean of \(W\): \(18.2\), Standard deviation of \(W\): \(\approx10.23\)
d. Mean of \(X\): \(69.6\), Standard deviation of \(X\): \(15.8\)