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Statistics

Center of a data set: mean, median, mode- Home
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- Discrete Probabilities

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Try reviewing these fundamentals first.

Statistics

Center of a data set: mean, median, modeStill Confused?

Try reviewing these fundamentals first.

Statistics

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Basic concepts: Center of a data set: mean, median, mode,

We can write out mean as an expected value:

$\mu=E[X]$

And likewise for variance:

$\sigma^2=$ Var$(X)$

*n: number of trials*

*x: number of success in n trials*

*p: probability of success in each trial*

**Binomial:**

$E[X]=np$

Var$(X)=np(1-p)$

**Geometric:**

$E[X]=\frac{1}{p}$

Var$(X)=\frac{1-p}{p^2}$

__Properties of Expectation:__

$\cdot$ $E[X+a]=E[X]+a$

$\cdot$ $E[bX]=bE[X]$

$\cdot$ $E[X+Y]=E[X]+E[Y]$

Or in full generality:

$\cdot$ $E[X_1+X_2+ \cdots +X_n ]=E[X_1 ]+E[X_2 ]+ \cdots +E[X_n]$

__Properties of Variance:__

$\cdot$ Var$[X+a]=$ Var$[X]$

$\cdot$ Var$[bX]=b^2$Var$[X]$

$\cdot$ Var$[X+Y]=$ Var$[X]+$ Var$[Y]$ if X and Y are independent

$\mu=E[X]$

And likewise for variance:

$\sigma^2=$ Var$(X)$

$E[X]=np$

Var$(X)=np(1-p)$

$E[X]=\frac{1}{p}$

Var$(X)=\frac{1-p}{p^2}$

$\cdot$ $E[X+a]=E[X]+a$

$\cdot$ $E[bX]=bE[X]$

$\cdot$ $E[X+Y]=E[X]+E[Y]$

Or in full generality:

$\cdot$ $E[X_1+X_2+ \cdots +X_n ]=E[X_1 ]+E[X_2 ]+ \cdots +E[X_n]$

$\cdot$ Var$[X+a]=$ Var$[X]$

$\cdot$ Var$[bX]=b^2$Var$[X]$

$\cdot$ Var$[X+Y]=$ Var$[X]+$ Var$[Y]$ if X and Y are independent

- 1.a)What is the expected value and variance for random variables?b)Properties of Expectation and Variance
- 2.
**Verifying Expectation and Variance**

If a 6 sided die is rolled what is the expected value shown on the die? - 3.A certain car breaks down every 50 hours of driving time. If the car is driven for a total of 175 hours;a)What is the expected number of breakdowns?b)What is the variance of the breakdowns?
- 4.Clara is trying to make the perfect teapot out of pottery. Each time she attempts to make the perfect teapot she will use a lump of clay and she will succeed with a probability of 0.20. Once she makes the perfect teapot she will stop potting.a)What is the expected number of lumps of clay Clara will use to make this perfect teapot?b)What is the variance on the number of lumps of clay Clara will use?
- 5.
**Properties of Expectation and Variance**

If a 6 sided die is rolled what is the expected value shown on the die? What would be the expected value if 10 die were rolled? - 6.Suppose we have two independent random variable one with parameters $E[X]=4$ and Var$(X)=3$, and the other with parameters $E[Y]=9$ and Var$(Y)=6$.a)What is $E[X+Y+2]$?b)What is $E[3X+2Y-5]$?c)What is Var$(3X+2)$?d)What is Var$(2(X+Y+1))$?

4.

Discrete Probabilities

4.1

Probability distribution - histogram, mean, variance & standard deviation

4.2

Binomial distribution

4.3

Mean and standard deviation of binomial distribution

4.4

Poisson distribution

4.5

Geometric distribution

4.6

Negative binomial distribution

4.7

Hypergeometric distribution

4.8

Properties of expectation

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