Expected value & variance of a discrete random variable, Study notes of Mathematics

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illustrate the mean and variance of a discrete

random variable

calculate the mean and the variance of a

discrete random variable

interpret the mean and the variance of a

discrete random variable

solve problems involving mean and variance of

probability distributions

1.) A security guard recorded the number of people

entering the bank every hour during one working day. The

random variable X represents the number of people who

entered the bank. The probability distribution of X is shown

below:

X 0 1 2 3 4 5 P(x) 0 0.1 0.2 0.4 0.2 0. What are the mass points of the distribution in tabular form?

What is the 𝐸 𝑥 of the distribution? 𝐸 𝑥^ = 3

2.) The organizing committee of a high school

reunion placed 150 balls inside a box. Ten of

which are red, five are blue, one is gold and the

rest are white. A player has a chance to draw

one ball. A red ball will win him ₱500, a blue ball

will win him ₱1,000 and the single gold ball can

give him ₱5,000. However, he won’t win any if

he draws a white ball. What would be the fair

price to pay for a chance to draw a ball form the

box?

a measure of dispersion (about the mean)

denoted by 𝜎^2 𝑜𝑟 𝑉𝑎𝑟 𝑋

implies that the larger the value of the variance,

the farther are the values of X from the mean,

in terms of formula:

𝜎^2 = 𝑥^2 ∙ 𝑃(𝑥) − ,E(x)-^2

Five marbles numbered 0,2,4,6 and 8 are placed

in a bag. After the balls are mixed, one ball is

picked and its number is noted, and then it is

replaced in the bag. If this experiment is

repeated many times, find the variance and

standard deviation of the numbers on the balls.

Variance: 𝜎^2 = 𝑉𝑎𝑟(𝑥) = 𝑥^2 ∙ 𝑃(𝑥) − ,E(x)-^2

number on ball (x) 0 2 4 6 8 P(x) 1 5

1 5

1 5

1 5

1 5

To find 𝜎^2 , solve: E(x) E x = 0 15 + 2 15 +4 15 +6 15 +8 (^15)

E x = 4

𝜎^2 = (0)^2

2

𝜎^2 = 8 Standard Deviation: 𝜎 = 𝑆𝐷 = 8 = 2.

Should the radio station consider more phone

lines installed?

Variance: 𝜎^2 = 𝑉𝑎𝑟(𝑥) = 𝑥^2 ∙ 𝑃(𝑥) − ,E(x)-^2

To find 𝜎^2 , solve: E(x) (^) E x = 1.

𝜎^2 = (0)^2 0.17 + (1)^2 0.34 + (2)^2 0.24 + (3)^2 0.20 + (4)^2 0.05 − ,1.62-^2 𝜎^2 = 1.

Standard Deviation: 𝜎 = 𝑆𝐷 = 1.28 = 1.

Since E x = 1.62, the station should not consider having more phone lines installed.

Juan tosses an unbiased coin. He receives ₱

if a head appears and he pays ₱30 if a tail

appears. Find the expected value and standard

deviation of his gain/loss.

X 50 -

P(x) 0.5 0.

Let X represent the number of times a student visits a nearby pizza parlor in a 1-month period. Assume that the following formula represents the probability distribution of X. Construct the probability distribution in tabular form.

𝑃(𝑥)

1 7 𝑥

(^2) , 𝑖𝑓 𝑥 = 0,1, 2 7 𝑥 − 2 , 𝑖𝑓 𝑥 = 3 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

a. Find μ and 𝜎. b. What is the probability that a student visits at least twice a month?

fin