Question

random variables
you just found that the mean of this discrete random variable is 7.96. what is the standard deviation of this discrete random variable?
2.82
3.33
63.36
9.07

Explanation:

Step1: Calculate variance formula

The formula for the variance \(\sigma^{2}=\sum(x - \mu)^{2}P(x)\), where \(\mu = 7.96\) is the mean.

Step2: Calculate \((x-\mu)^{2}P(x)\) for each \(x\)

• When \(x = 4\): \((4 - 7.96)^{2}\times0.20=( - 3.96)^{2}\times0.20 = 15.6816\times0.20=3.13632\)
• When \(x = 6\): \((6 - 7.96)^{2}\times0.34=( - 1.96)^{2}\times0.34 = 3.8416\times0.34 = 1.306144\)
• When \(x = 9\): \((9 - 7.96)^{2}\times0.15=(1.04)^{2}\times0.15 = 1.0816\times0.15=0.16224\)
• When \(x = 11\): \((11 - 7.96)^{2}\times0.22=(3.04)^{2}\times0.22 = 9.2416\times0.22 = 2.033152\)
• When \(x = 15\): \((15 - 7.96)^{2}\times0.09=(7.04)^{2}\times0.09 = 49.5616\times0.09 = 4.460544\)

Step3: Sum up the values

\(\sigma^{2}=3.13632 + 1.306144+0.16224 + 2.033152+4.460544 = 11.0984\)

Step4: Calculate standard deviation

The standard deviation \(\sigma=\sqrt{\sigma^{2}}\), so \(\sigma=\sqrt{11.0984}\approx3.33\)

Answer:

3.33