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# 12. Variance and Standard Deviation

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The following topics are included in this series of six videos.

1. Introduction to variance and standard deviation
2. Variance and standard deviation, example 1
3. Variance and standard deviation, example 2, involves completing a pdf first
4. Variance and standard deviation, example 3 (part 1), involves combinations
5. Variance and standard deviation, example 3 (part 2), involves combinations
6. Variance and standard deviation, example 4, shortcut for Bernoulli processes

#### Prework

1. Consider the random variable $$X$$ and its pdf given in the table below. Determine the expected value, variance, and standard deviation of $$X$$.
 $$X$$ Probability 10 .1 20 .5 30 .4
2. Consider the random variable $$Y$$ and its pdf given in the table below. Determine $$E[Y]$$, $$Var[Y]$$, and $$\sigma$$ for $$Y$$.

 $$Y$$ Probability 8 $$\frac{1}{2}$$ 12 $$\frac{1}{3}$$ 24 $$\frac{1}{6}$$
3. A softball player gets a hit 30% of the time, and fails to get a hit 70% of the time. Suppose she goes up to bat 1000 times in her career. What is the expected number of hits she gets? What is the variance and standard deviation of the number of hits she gets?

1.  $$X$$ Probability Product $$(X-E[X])^2$$ Product 10 .1 1 169 16.9 20 .5 10 9 4.5 30 .4 12 49 19.6 $$E[X]=23$$ $$Var(X)=41, \sigma=\sqrt{41}$$
2.  $$Y$$ Probability Product $$(Y-E[Y])^2$$ Product 8 $$\frac{1}{2}$$ 4 16 8 12 $$\frac{1}{3}$$ 4 0 0 24 $$\frac{1}{6}$$ 4 144 24 $$E[Y]=12$$ $$Var(Y)=32, \sigma=\sqrt{32}$$
3. If we let $$X=$$the number of hits she gets in 1000 attempts, then we are in a situation in which we can use the Bernoulli shortcuts. Therefore $$E[X]=np=1000\cdot .3=300, Var(X)=np(1-p)=1000\cdot .3\cdot .7=210,$$ and $$\sigma=\sqrt{210}$$.