Discrete and Continuous

Harold Nelson

2025-07-21

Intro

I’m going to use a few utility functions in R to clarify the difference between discrete and continuous random variables.

Discrete

For a discrete example I’ll use a binomial random variable with a success probability of .5 and four trials. Use the rbinom() function to get 100 simulated values.

Solution

binom_values = rbinom(100,4,.5)

Look at them

Solution

binom_values

##   [1] 1 1 1 2 3 0 4 1 1 2 1 2 0 3 0 1 2 1 4 3 3 2 2 1 1 1 3 1 2 1 2 1 2 3 1 3 0
##  [38] 2 0 1 4 3 2 1 4 4 3 3 2 2 3 3 3 1 2 3 2 1 4 2 2 2 1 0 2 3 3 1 3 3 2 2 1 1
##  [75] 0 2 2 1 3 2 1 2 1 1 2 2 4 2 2 3 1 1 4 1 3 2 3 2 2 2

Get a histogram

Solution

hist(binom_values)

Get a table

Solution

binom_table = table(binom_values)
binom_table

## binom_values
##  0  1  2  3  4 
##  7 30 33 22  8

Probabilities

Divide the numbers in the table to probabilities by dividing by 100.

Solution

binom_probs = binom_table/100
binom_probs

## binom_values
##    0    1    2    3    4 
## 0.07 0.30 0.33 0.22 0.08

Check

Do the probabilities add up to 1?

Solution

sum(binom_probs)

## [1] 1

Yes.

Continuous

Create 100 simulated values from a normal distribution with the same mean and standard deviation as the binomial.

Solution

binom_mean = mean(binom_values)
binom_sd = sd(binom_values)

binom_mean

## [1] 1.94

binom_sd

## [1] 1.061921

## The Simulated Normal Values

norm_values = rnorm(100,
                    mean = binom_mean,
                    sd = binom_sd)

Look

Solution

hist(norm_values)

summary(norm_values)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
## -0.3435  1.1770  2.0674  1.9237  2.5807  4.6347

The distribution looks very similar to the binomial.

Table and Probabilities

This is the punch line. Try to repeat what we did with the binomial.

Solution

norm_table = table(norm_values)
norm_table

## norm_values
##  -0.343491984414252  -0.258880367359692   -0.16095863694362 -0.0228714596410502 
##                   1                   1                   1                   1 
## -0.0132547216066157   0.176573700916938   0.202393823791152   0.341611237715347 
##                   1                   1                   1                   1 
##   0.404077297622685   0.421105625156222   0.462808227714427   0.527014947101507 
##                   1                   1                   1                   1 
##   0.815517048991347   0.866168428795573   0.905064423210082   0.909887567686061 
##                   1                   1                   1                   1 
##   0.977260641913571    1.00951410401927    1.01385275855134    1.08091424641425 
##                   1                   1                   1                   1 
##     1.0897810115423     1.1119199158189    1.13082793877382    1.14256756698486 
##                   1                   1                   1                   1 
##    1.14265155712979    1.18839418853991    1.21880254806153    1.28237574841731 
##                   1                   1                   1                   1 
##    1.28878274993702    1.29350138053226    1.30508556109448    1.31603514801407 
##                   1                   1                   1                   1 
##    1.39849294879992    1.40868444827741    1.41650278057396    1.44310825069849 
##                   1                   1                   1                   1 
##    1.46669532543228    1.47816462822507      1.495128646612    1.56287626894939 
##                   1                   1                   1                   1 
##    1.70501481177497    1.71524784961572    1.74113529474139    1.78851938387562 
##                   1                   1                   1                   1 
##    1.84948323730858    1.91111530471791    1.92115318033495    2.03314956875144 
##                   1                   1                   1                   1 
##    2.04575460235279    2.06585684538196     2.0688762080369    2.08329976716188 
##                   1                   1                   1                   1 
##    2.12752579689503    2.14427552960416    2.15217140218523    2.16574478625171 
##                   1                   1                   1                   1 
##    2.16773080047055    2.19425351240454    2.20569802476601    2.22117722292526 
##                   1                   1                   1                   1 
##    2.25344153466385    2.31096731802082    2.31803724992206    2.33234218964121 
##                   1                   1                   1                   1 
##    2.33452190451542    2.35094419536478    2.43661643042227    2.45515703822999 
##                   1                   1                   1                   1 
##    2.46298388649132    2.46514290959299    2.50067746822742    2.50847817031168 
##                   1                   1                   1                   1 
##      2.522538618447     2.5443243850241    2.56579086672253    2.62554018519417 
##                   1                   1                   1                   1 
##    2.64301668106617    2.65165586054842    2.70507027002433    2.75758456715313 
##                   1                   1                   1                   1 
##    2.75952082934516     2.7760067524127    2.92250861235869    2.98403388669574 
##                   1                   1                   1                   1 
##    2.98465835218072    2.98578704204083    2.99460873041245    2.99502739568719 
##                   1                   1                   1                   1 
##    3.01878820861603    3.08148342956341    3.12410711488919    3.33498807958625 
##                   1                   1                   1                   1 
##    3.39248236342935    3.50491812485095    3.69756276866098     3.7628034935207 
##                   1                   1                   1                   1 
##    3.83565022928607    3.84750450133603    4.18344590887875    4.63469877803152 
##                   1                   1                   1                   1

Every value occurs once. When we convert these to probabilities, they are all the same - .01.

Do It

Solutiion

norm_probs = norm_table/100
norm_probs

## norm_values
##  -0.343491984414252  -0.258880367359692   -0.16095863694362 -0.0228714596410502 
##                0.01                0.01                0.01                0.01 
## -0.0132547216066157   0.176573700916938   0.202393823791152   0.341611237715347 
##                0.01                0.01                0.01                0.01 
##   0.404077297622685   0.421105625156222   0.462808227714427   0.527014947101507 
##                0.01                0.01                0.01                0.01 
##   0.815517048991347   0.866168428795573   0.905064423210082   0.909887567686061 
##                0.01                0.01                0.01                0.01 
##   0.977260641913571    1.00951410401927    1.01385275855134    1.08091424641425 
##                0.01                0.01                0.01                0.01 
##     1.0897810115423     1.1119199158189    1.13082793877382    1.14256756698486 
##                0.01                0.01                0.01                0.01 
##    1.14265155712979    1.18839418853991    1.21880254806153    1.28237574841731 
##                0.01                0.01                0.01                0.01 
##    1.28878274993702    1.29350138053226    1.30508556109448    1.31603514801407 
##                0.01                0.01                0.01                0.01 
##    1.39849294879992    1.40868444827741    1.41650278057396    1.44310825069849 
##                0.01                0.01                0.01                0.01 
##    1.46669532543228    1.47816462822507      1.495128646612    1.56287626894939 
##                0.01                0.01                0.01                0.01 
##    1.70501481177497    1.71524784961572    1.74113529474139    1.78851938387562 
##                0.01                0.01                0.01                0.01 
##    1.84948323730858    1.91111530471791    1.92115318033495    2.03314956875144 
##                0.01                0.01                0.01                0.01 
##    2.04575460235279    2.06585684538196     2.0688762080369    2.08329976716188 
##                0.01                0.01                0.01                0.01 
##    2.12752579689503    2.14427552960416    2.15217140218523    2.16574478625171 
##                0.01                0.01                0.01                0.01 
##    2.16773080047055    2.19425351240454    2.20569802476601    2.22117722292526 
##                0.01                0.01                0.01                0.01 
##    2.25344153466385    2.31096731802082    2.31803724992206    2.33234218964121 
##                0.01                0.01                0.01                0.01 
##    2.33452190451542    2.35094419536478    2.43661643042227    2.45515703822999 
##                0.01                0.01                0.01                0.01 
##    2.46298388649132    2.46514290959299    2.50067746822742    2.50847817031168 
##                0.01                0.01                0.01                0.01 
##      2.522538618447     2.5443243850241    2.56579086672253    2.62554018519417 
##                0.01                0.01                0.01                0.01 
##    2.64301668106617    2.65165586054842    2.70507027002433    2.75758456715313 
##                0.01                0.01                0.01                0.01 
##    2.75952082934516     2.7760067524127    2.92250861235869    2.98403388669574 
##                0.01                0.01                0.01                0.01 
##    2.98465835218072    2.98578704204083    2.99460873041245    2.99502739568719 
##                0.01                0.01                0.01                0.01 
##    3.01878820861603    3.08148342956341    3.12410711488919    3.33498807958625 
##                0.01                0.01                0.01                0.01 
##    3.39248236342935    3.50491812485095    3.69756276866098     3.7628034935207 
##                0.01                0.01                0.01                0.01 
##    3.83565022928607    3.84750450133603    4.18344590887875    4.63469877803152 
##                0.01                0.01                0.01                0.01

This is true, but there is something wrong. It says that a particular value near 2 is no more likely to occur than a particular value near 0 or 4.

This was with 100 values. If we’d used 10,000, every individual value would have a probability of .0001. As the size of our sample approaches infinity, the probability of any particular value approaches 0.

What’s the punch line.

Solution

With a continuous random variable, you have to attach probabilities to intervals, not individual values.

A Specific Solution

We can use a trick. The mean value of a logical expression is the fraction of cases for which the logical expression is true. What happens when you do this? The value TRUE is coerced to 1 and the value FALSE is coerced to 0.

Less than 4

prob_lt_4 = mean(norm_values < 4)
prob_lt_4

## [1] 0.98

Less than 3

prob_lt_3 = mean(norm_values < 3)
prob_lt_3

## [1] 0.88

Between 3 and 4

prob_between_3_4 = prob_lt_4 - prob_lt_3
prob_between_3_4

## [1] 0.1

Between 1 and 2

prob_between_1_2 = mean(norm_values < 2) - mean(norm_values < 1)
prob_between_1_2

## [1] 0.3

The Bottom Line

Discrete: Individual Values.
Continuous: Intervals