CABT SHS Statistics & Probability - Expected Value and Variance of Discrete Probability Distributions

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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)


Remarks:1. “The mean of the

random variable” can also mean “the mean of the probability distribution.”



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.


Find the mean of the

given probability

distribution.


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 of a Random Variable2 Solution

Recall that the pdf for the number of heads is as follows:


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


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.


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 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?



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?



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.






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




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.






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




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!

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Education

CABT SHS Statistics & Probability - Expected Value and Variance of Discrete Probability Distributions