MA 115, Fall 2011
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Contents
Distribution Of The Sample Mean
Distribution Of The Sample Mean
Example
A professional Hold'Em player claims to win 1 big blind per 1 hour session with standard deviation of 12 big blinds per hour. If the winnings from different sessions are independent, find the probability that the average amount won after 30 hours is positive. Do the same for 500 hours.
We can regard the winnings in sessions as independent identically distributed (iid) random variables , , ... , . Since the size of this sample is pretty big, we can rely on the Central Limit Theorem. Regardless of the distribution of each , we know that the distribution of the average winnings is approximately normal with mean and standard deviation
.
And so the answer is
or approximately . Since , we can conclude that after playing 30 hours, the player comes out in black about of the time.
Similarly, if then
and
or approximately . As a consequence, there is a whopping chance that the player will still be in the red after playing for 500 hours.
Expected Value And Variance
Expected Value, Discrete Case
For a discrete random variable with a corresponding probability mass function we define the expected value of (denoted by ) to be the sum
For example, let be the amount of money (in USD) you earn in a lottery where you buy a 1 USD ticket and have a small chance to win either a 100 or 10 USD prize. Let , , and . That is, there is a large chance that you win no prize but still have to pay 1 USD for the ticket. Here's the pmf of in table form (note that probabilities add up to ):
Then
That is, every time you play this lottery, you lose 20 cents on average.
Variance
The variance of any random variable (denoted ) is defined to be
.
A basic theorem about variance gives us another way of expressing it:
.
To find the variance of a discrete random variable, we can construct a pmf for and compute . Using the variable from the example above, we have
And so
and hence
.
This is rather large variance (much larger than the average value of ), just as one would expect from a lottery. You lose a few cents on average, but there is a distinct possibility of earning many dollars every once in a while.
Chebyshev's Inequality
When applied to random variables, Chebyshev's inequality can be stated as follows: for any random variable with mean and variance and any real number , at least of the possible values of are within standard deviations of the mean. In terms of probability,
The inequality can be stated for any particular value of . For example, if , then at least of possible values are within standard deviations of the mean; if , then at least of the possible values are within standard deviations of the mean, and so on.
Example
The height of a hobbit in his or her tweens follows the distribution with mean inches and standard deviation inches. Use Chebyshev's inequality to (a) put a bound on the proportion of hobbits whose height is between 30 and 42 inches and (b) find the interval which is guaranteed to contain 95% of the population.
(a) We need the proportion of the population within of the mean, so we can use the inequality directly for ,
.
(b) We can find which corresponds to the interval which covers 95% of the population by solving the equation . Since , the interval can be written as , or approximately .
Stem And Leaf Plot Examples
Example 1
Sorted data:
43 | 45 | 47 | 49 | 62 | 66 | 67 | 69 | 72 | 72 |
75 | 76 | 81 | 84 | 88 | 104 | 104 | 105 | 105 | 105 |
Stem and Leaf plot:
The decimal point is 1 digit(s) to the right of the | 4 | 3579 5 | 6 | 2679 7 | 2256 8 | 148 9 | 10 | 44555
Example 2
Sorted data:
-2.300 | -1.500 | -1.200 | -1.100 | -0.790 | -0.580 | -0.550 | -0.260 | -0.190 | -0.130 |
-0.072 | 0.210 | 0.380 | 0.800 | 0.950 | 0.990 | 1.300 | 1.400 | 1.400 | 1.500 |
Stem and Leaf plot (rounding away from zero)
The decimal point is at the | -2 | 3 -1 | 521 -0 | 8663211 0 | 248 1 | 003445
See here for more info.