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Mean and Variance of Discrete Probability
Distributions
Statistics & ProbabilityGrade 11
Mean and Variance of Random Variables
For a discrete random variable X with probability distribution function P(X), the EXPECTED
VALUE or MEAN of X is given by
E X xP x
Mean of a Random Variable
The mean is equal to the SUM of the PRODUCTS of the values of X and their corresponding
probabilities P(X)
Mean and Variance of Random Variables
Remarks:1. “The mean of the
random variable” can also mean “the mean of the probability distribution.”
Mean of a Random Variable
Mean and Variance of Random Variables
Remarks:2. The mean of a random
variable is the weighted AVERAGE value of all the possible values of X, with the weights being the probabilities of the respective values of X.
Mean of a Random Variable
Find the mean of the
given probability
distribution.
Mean and Variance of Random Variables
Mean of a Random Variable1 X P(X)
0 0.051 0.152 0.353 0.45
number of
head X0 1 2 3
P(X) 1/8 3/8 3/8 1/8
If three coins are
tossed, find the mean of the number of heads X that occur.
Mean and Variance of Random Variables
Mean of a Random Variable2 Solution
Recall that the pdf for the number of heads is as follows:
Mean and Variance of Random Variables
For a discrete random variable X with probability distribution function P(X),
the VARIANCE of X is given by 2 2 2Var X x P x
Variance of a Random Variable
Note: so 2 2x P x E X 2 2 2E X
Mean and Variance of Random Variables
The STANDARD DEVIATION of a discrete random
variable X is the SQUARE ROOT of its variance:
2 2x P x
The Standard Deviation
X P(X)0 0.051 0.152 0.353 0.45
Find the variance and
standard deviation of the given probability
distribution.
Mean and Variance of Random Variables
Variance of a Random Variable3
A study conducted by a TV station showed the number of televisions per household and the corresponding probabilities for each. Find the mean, variance, and standard deviation.
Mean and Variance of Random Variables
Mean of a Random Variable4
a.An investment in Project A will result in a loss of P 26,000 with probability 0.30, break even with probability 0.50, or result in a profit of P 68,000 with probability 0.20. What is the expected value of this investment?
Mean and Variance of Random Variables
Mean of a Random Variable5
b. Meanwhile, an investment in Project B will result in a loss of P 71,000 with probability 0.20, break even with probability 0.65, or result in a profit of P 143,000 with probability 0.15. Which investment is better: A or B?
Mean and Variance of Random Variables
Mean of a Random Variable5
In the Mr. and Ms. CABT Raffle Draw, there is one winner of P 10,000, one winner of P 5000, and five winners of P1000. One thousand tickets are sold at P10 each. Find the expectation if a person buys one ticket.
Mean and Variance of Random Variables
Mean of a Random Variable6
In the Mr. and Ms. CABT Raffle Draw, there is one winner of P 10,000, one winner of P 5000, and five winners of P1000. One thousand tickets are sold at P10 each. Find the expectation if a person buys one ticket.
Mean and Variance of Random Variables
Mean of a Random Variable
6Prize 0 1,000 5,000 10,00
0Gain (X) 10 990 4,990 9,990No. of tickets 993 5 1 1Probabilit
y 0.993 0.005 0.001 0.001
In the Mr. and Ms. CABT Raffle Draw, there is one winner of P 10,000, one winner of P 5000, and five winners of P1000. One thousand tickets are sold at P10 each. Find the expectation if a person buys one ticket.
Mean and Variance of Random Variables
Mean of a Random Variable
6
Gain X 10 990 4,990 9,990
P(X) 0.993 0.005 0.001 0.001
The following is the pdf:
JM the financial adviser suggests that his client Manay Ehljie select one of two types of bonds in which to invest her P 5,000. Bond X pays a return of 4% and has a default rate of 2%. Bond Y has a return of 2.5% and a default rate of 1%. Find the expected rate of return and decide which bond would be a better investment. When the bond defaults, the investor loses all the investment.
Mean and Variance of Random Variables
Mean of a Random Variable7
JM the financial adviser suggests that his client Manay Ehljie select one of two types of bonds in which to invest her P 5,000. Bond X pays a return of 4% and has a default rate of 2%. Bond Y has a return of 2.5% and a default rate of 1%. Find the expected rate of return and decide which bond would be a better investment. When the bond defaults, the investor loses all the investment.
Mean and Variance of Random Variables
Mean of a Random Variable
7Amount of return for X:
5,000 0.04 200Amount of return for Y:
5,000 0.025 125The probability of obtaining a return
(that is, NOT defaulting):For X: 100% 2% = 98% or 0.98
For Y: 100% 2.5% = 97% or 0.975
JM the financial adviser suggests that his client Manay Ehljie select one of two types of bonds in which to invest her P 5,000. Bond X pays a return of 4% and has a default rate of 2%. Bond Y has a return of 2.5% and a default rate of 1%. Find the expected rate of return and decide which bond would be a better investment. When the bond defaults, the investor loses all the investment.
Mean and Variance of Random Variables
Mean of a Random Variable
7Return X 5,000 200
P(X) 0.02 0.98
Return Y 5,000 125P(Y) 0.01 0.975
Do you have any QUESTIONs?
Summing it up!
When I listen, I hear.
When I see, I remember.
When I do, I understand.
Final Words….
Thank you!