Mean of a Random Variable
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Mean of a Random Variable uppose that X is a discrete random variable whose istribution is as follows : Value of X Probability x 1 x 2 x 3 ... x k p 1 p 2 p 3 p k ... o find the mean of X, multiply each possible value ts probability, then add all the products. X = x 1 p 1 + x 2 p 2 + x 3 p 3 + … + x k p k = x i p i
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Mean of a Random Variable. x 1. x 2. x 3. x k. Value of X. p 1. p 2. p 3. p k. Probability. Suppose that X is a discrete random variable whose distribution is as follows :. To find the mean of X, multiply each possible value by its probability, then add all the products. - PowerPoint PPT Presentation
Transcript of Mean of a Random Variable
Mean of a Random Variable
Suppose that X is a discrete random variable whosedistribution is as follows :
Value of X
Probability
x1 x2 x3 ... xk
p1 p2 p3 pk...
To find the mean of X, multiply each possible value byits probability, then add all the products.
X = x1p1 + x2p2 + x3p3 + … + xkpk
= xipi
Mean of a Random VariableExample : Nelson is trying to get Homer to play a game
of chance. This game costs $2 to play. The game is toflip a coin three times, and for each head that appears,Homer will win $1.
Here is the probability distribution :
Number of Heads
Probability
0 1 2 3
.125 .375 .375 .125
What is the expected payoff of this game? The mean!
Mean of a Random Variable
Number of Heads
Probability
0 1 2 3
.125 .375 .375 .125
X = (0)(.125) +(1)(.375) +(2)(.375) +(3)(.125) =1.5
In this game, Homer would expect to win $1.50
So, should Homer play this game ?No.
Rules for Means1) If X is a random variable and a and b are fixed numbers, then :
a+bX = a + bX
Example : What if Homer tries Nelsons game, but he decides to award $10 for every head that Nelson flips.
Number of Heads
Probability
0 10 20 30
.125 .375 .375 .125
X = (0)(.125) + (10)(.375) + (20)(.375) + (30)(.125) = 15
Rules for Means1) If X is a random variable and a and b are fixed numbers, then :
a+bX = a + bX
Example : What if Homer tries Nelsons game, but he decides to award $10 for every head that Nelson flips.
Number of Heads
Probability
0 10 20 30
.125 .375 .375 .125
Notice that this is just a linear transformation withb = 10 and a = 0.
10X = a + bX = 0 + 10X = (10)(1.5) = 15
(Same as before)
Rules for Means
2) If X and Y are random variables, then
X+Y = X + Y
Example :
Let X be a random variable with X = 12
Let Y be a random variable with Y = 18
What is the mean of X + Y ?
X+Y = X + Y = 12 + 18 = 30
Variance of a Discrete Random Variable
Suppose that X is a discrete random variable whosedistribution is :
Value of X
Probability
x1 x2 x3 ... xk
p1 p2 p3 pk...
and that X is the mean of X. The variance of X is :
X
2= (x1 - X)2 p1 + (x2 - X)2 p2 + ... + (xk - X)2 pk
= (xi - X)2 pi
The standard deviation X of X is the square root of the variance.
Variance of a Discrete Random Variable
Example : Find the variance of Nelson’s game.
Number of Heads
Probability
0 1 2 3
.125 .375 .375 .125
Recall the mean was 1.5
X
2= (x1 - X)2 p1 + (x2 - X)2 p2 + ... + (xk - X)2 pk
= (0 - 1.5)2 (.125) +(1 - 1.5)2 (.375) +(2 - 1.5)2 (.375) +
(3 - 1.5)2 (.125)
= .28125 + .09375 + .09375 + .28125 =0.75
X = 0.75 = .8660254
Rules for Variances
1) If X is a random variable and a and b are fixed numbers, then :
a + bX
2 X
2= b2
2) If X and Y are independent random variables, then
X + Y
2 X
2 Y
2= +
X - Y
2 X
2 Y
2= +
Rules for Variances
Example : Mike’s golf score varies from round to round.
X = 88 X = 8
Tiger’s golf score vary from round to round.
Y = 92 Y = 9
Q: What is the expected difference between our two rounds ?
X-Y = X - Y
= 88 - 92 = -4
Rules for Variances
Example : Mike’s golf score varies from round to round.
X = 88 X = 8
Tiger’s golf score vary from round to round.
Y = 92 Y = 9
Q: What is the variance of the differences ?
X - Y
2 X
2 Y
2= +
= 64 + 81 = 145
X-Y = 145 = 12.041594
The Law of Large Numbers
• Draw independent observations at random from any population with finite mean
• Decide how accurately you want to estimate
• As the number of observations increases, the mean of the observed values eventually approaches the mean of the population.
Example:
M & M’s• Brown 30%• Red 20%• Yellow 20%
• Blue 10%• Green 10%• Orange 10%
M & M’s• Brown 30%• Red 20%• Yellow 20%
• Blue 10%• Green 10%• Orange 10%
Law of Large Numbers
During an experiment, as the number of experiments grows larger, then the observed mean will grow closerand closer to the actual mean.
M & M’s• Brown 30%• Red 20%• Yellow 20%
• Blue 10%• Green 10%• Orange 10%
Red Brown Yellow Blue Green Orange TotalsRed #DIV/0! 20Brown #DIV/0! 30Yellow #DIV/0! 20Blue #DIV/0! 10Green #DIV/0! 10Orange #DIV/0! 10
Totals 0 0 0 0 0 0
Homework50, 51, 54, 55, 56, 58, 60, 61, 62, 66,69
