6.1 Discrete and Continuous Random Variables, Exercises of Signals and Systems

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Discrete and

Continuous Random

Variables

6.1A

Discrete random

Variables, Mean

(Expected Value) of a

Discrete Random

Variable

Random variable Takes numerical values that describe the

outcomes of some chance process

Probability distribution Describes the possible values a

variable can take and how often it takes those values.

Number of children in a family.

The Friday night attendance at a cinema.

The number of patients in a doctor's surgery.

The number of defective light bulbs in a box of ten.

Suppose we flip a coin 3 times and the sample space is as follows:

S = {HHH, HHT, HTH, THH, HTT, THT, TTH, TTT}

Define the random variable X = the number of heads obtained

b) Make a histogram of the probability distribution. Describe what you

see.

c) Describe P(X ≥

6. in words and find its value?

P ( X

The probability that a randomly selected team scored at least 6 goals

is 0.061.

d) What is the probability that a randomly selected team scores more

than 6 goals in a game?

P ( X> 6) = 0.015 + 0.004 + 0.001 = 0.

Mean (Expected Value) of a Discrete random variable

“What can we expect in the long run?”

It is an average of the possible outcomes, but a weighted average in

which each outcome is weighted by its probability

Included on the AP Exam formula sheet

The expected value may NOT be equal to one of the possible values of

the variable

Variance and standard deviation of a Discrete random variable

How much the values of the variable tend to vary, on average, from the

expected value. The average distance the outcomes are from the mean.

Example

A wager that players can make in roulette is called a “corner bet.” To make

this bet, a player places his chips on the intersection of four numbered

squares on the roulette table. If one of these numbers comes up on the wheel

and the player bets $1, the player gets his $1 back plus $8 more. Otherwise,

the casino keeps the original $1 bet. If X = the net gain from a single $

corner bet, the possible outcomes are X = 1 or X = 8. Here is the probability

distribution of X for 38 bets.

Value: 1 8

Probability: 34/38 4/

a) What is the player’s average gain?

Example

Refer to Example #1(NHL) and compute the mean and the standard

deviation of the random variable X and interpret these values in context.

(b) Find the probability that a randomly selected threeyearold female

weighs between 25 and 35 pounds.

Example

Joe the barber charges $32 for a shave and haircut and $20 for just a haircut.

Based on experience, he determines that the probability that a randomly

selected customer comes in for a shave and haircut is 0.85, the rest of his

customers come in for just a haircut. Let J = what Joe charges a randomly

selected customer.

(a) Give the probability distribution for J.

J 32 20

P(J) 0.85 0.

(b) Find and interpret the mean of J.

(c) Find and interpret the standard deviation of J.