6.1 Random Variables

Random variable – variable whose value is a numerical outcome of a random phenomenon.

Discrete – countable number of number of probable outcomes

Continuous – measurable ex. Temperature, height, age

Discrete variables

• Discrete example

Cars owned

x

Number of

families  

relative frequency or P(x) xP(x)

0 17 0.00425 0

1 877 0.21925 0.219

2 1767 0.44175  

3 1107 0.27675 0.83

4 200 0.05  

5 32 0.008  

  4000   1  

A) Find P(x=0)

B) Find P(x≤2)

C) Find P(x >2)

D) Find P(x≥1)

Random variables

• Population mean μ also called “expected value”• μ = Σ xP(x)

• variance• σ2 = Σ (x- μ)2P(x)

• Standard deviation = σ• Find μ and σ for last problem

Continuous variablesWhat was the age distribution of nurses in Great Britain at

the time of Florence Nightingale 1851 25466 nurses in Great Britain?

Age Midpoint P(x) xP(x) x-μ (x-μ)2 (x-μ)2P(x)

20-29 24.5 0.057 1.4 -29.3 856 48.8

30-39 34.5 0.097 3.35 -19.3 371 36

40-49 44.5 0.197 8.77 -9.26 85.7 16.9

50-59 54.5 0.292 15.9 0.74 0.55 0.16

60-69 64.5 0.25 16.1 10.7 115 28.8

70-79 74.5 0.091 6.78 20.7 430 39.1

80+ 84.5 0.018 1.52 30.7 945 17

1.002 53.8 187 13.7

Continuous variables

A) Draw a histogram of the distribution

B) Find the probability that a nurse selected at random would be 60 or older.

C) What is the expected age of nurses in 1851?

D) Find the standard deviation.

Random variables

USA Today reported that about 25% of all state prisoners released on Parole become repeat offenders while on parole. Suppose the parole board is examining 5 prisoners for parole. Let x= number of prisoners of the 5 on parole who become repeat offenders.

Here is the probability distribution:

Random variables

x 0 1 2 3 4 5

P(x) 0.24 0.4 0.26 0.09 0.02 0

a) Find the probability that one or more of the 5 will be repeat offenders?

b) Find the probability that 2 or more will be repeat offenders?

c) Find the probability that 4 or more will be repeat offenders?

d) Find the expected number of repeat offenders?e) Find the standard deviation.