section 6.1 homework\ndue monday by 11:59pm points 21 submitting an external tool\nsection 6.1…
section 6.1 homework\ndue monday by 11:59pm points 21 submitting an external tool\nsection 6.1 homework\nscore: 7.67/21 answered: 8/21\nquestion 11\nconsider the discrete random variable x given in the table below. calculate the standard deviation of x. round answers to two decimal places.\n| x | 3 | 10 | 12 | 14 | 19 |\n| p(x) | 0.13 | 0.59 | 0.12 | 0.08 | 0.08 |\nμ = \nσ² = \nσ = \nwhat is the expected value of x?
Answer
Explanation:
Step1: Calculate the expected value $\mu$
The formula for the expected value of a discrete - random variable is $\mu=\sum_{i}x_ip_i$. $\mu = 3\times0.13+10\times0.59 + 12\times0.12+14\times0.08+19\times0.08$ $=0.39 + 5.9+1.44 + 1.12+1.52$ $=10.37$
Step2: Calculate the variance $\sigma^{2}$
The formula for the variance of a discrete - random variable is $\sigma^{2}=\sum_{i}(x_i-\mu)^2p_i$. $(3 - 10.37)^2\times0.13+(10 - 10.37)^2\times0.59+(12 - 10.37)^2\times0.12+(14 - 10.37)^2\times0.08+(19 - 10.37)^2\times0.08$ $=(-7.37)^2\times0.13+(-0.37)^2\times0.59+(1.63)^2\times0.12+(3.63)^2\times0.08+(8.63)^2\times0.08$ $=54.3169\times0.13 + 0.1369\times0.59+2.6569\times0.12 + 13.1769\times0.08+74.4769\times0.08$ $=7.061197+0.080771+0.318828+1.054152+5.958152$ $=14.473099\approx14.47$
Step3: Calculate the standard deviation $\sigma$
The standard deviation is the square - root of the variance, $\sigma=\sqrt{\sigma^{2}}$. $\sigma=\sqrt{14.47}\approx3.80$
Answer:
$\mu = 10.37$, $\sigma^{2}=14.47$, $\sigma = 3.80$