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Discrete and Continuous Random Variables: A Comprehensive Overview, Slides of Discrete Mathematics

Rice UniversityDiscrete Mathematics

A continuous random variable is characterized by its probability density function, a graph which has a total area of 1 beneath it: The probability of the random ...

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Discrete Random Variables
A dichotomous random variable takes only the values 0 and 1. Let X be such a random
variable, with Pr(X=1) = p and Pr(X=0) = 1-p . Then E[X] = p, and Var[X] = p(1-p) .
Consider a sequence of n independent experiments, each of which has probability p of “being
a success.” Let Xk = 1 if the k-th experiment is a success, and 0 otherwise. Then the total
number of successes in n trials is X = X1 +...+ Xn ; X is a binomial random variable, and
Pr(X=k) = n
kp(1 - p ) .
kn-k
⎛
⎝
⎜⎞
⎠
⎟⋅
E[X] = np , and Var[X] = np(1-p) . (These results follow from the properties of the expected
value and variance of sums of independent random variables.)
Next, consider a sequence of independent experiments, and let Y be the number of trials up to
(and including) the first success. Y is a geometric random variable, and
Pr(Y = k) = (1- p)p .
k-1
E[Y] = 1/p , and Var[Y] = (1-p)/p2 . (These results follow from the evaluation of infinite sums.)
A hypergeometric random variable Z results from drawing a sample of size n from a
population of size N containing g “good” members, and then counting the number of “good”
members in the sample:
.
n
N
zn
gN
z
g
= z)=Pr(Z
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
−
⋅
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
(This formula was used to compute the relevant probabilities in the “Bag R vs. Bag B” example.)
pf3
pf4
pf5

Partial preview of the text

Download Discrete and Continuous Random Variables: A Comprehensive Overview and more Slides Discrete Mathematics in PDF only on Docsity!

Discrete Random Variables

A dichotomous random variable takes only the values 0 and 1. Let X be such a random

variable, with Pr(X=1) = p and Pr(X=0) = 1-p. Then E[X] = p, and Var[X] = p(1-p).

Consider a sequence of n independent experiments, each of which has probability p of “being

a success.” Let Xk = 1 if the k-th experiment is a success, and 0 otherwise. Then the total

number of successes in n trials is X = X 1 +...+ Xn ; X is a binomial random variable , and

Pr(X = k) =

n

k

⎛ p (1 - p )k n- k.

⎝⎜^

E[X] = np , and Var[X] = np(1-p). (These results follow from the properties of the expected

value and variance of sums of independent random variables.)

Next, consider a sequence of independent experiments, and let Y be the number of trials up to

(and including) the first success. Y is a geometric random variable , and

Pr(Y = k) = (1- p ) k-1p.

E[Y] = 1/p , and Var[Y] = (1-p)/p^2. (These results follow from the evaluation of infinite sums.)

A hypergeometric random variable Z results from drawing a sample of size n from a

population of size N containing g “good” members, and then counting the number of “good”

members in the sample:

n

N

n z

N g

z

g

Pr(Z=z)=

(This formula was used to compute the relevant probabilities in the “Bag R vs. Bag B” example.)

Continuous Random Variables

A continuous random variable is a random variable which can take any value in some interval. A

continuous random variable is characterized by its probability density function , a graph which has

a total area of 1 beneath it: The probability of the random variable taking values in any interval

is simply the area under the curve over that interval.

The normal distribution : This most-familiar of continuous probability distributions has the

classic “bell” shape (see the left-hand graph below). The peak occurs at the mean of the

distribution, i.e., at the expected value of the normally-distributed random variable with this

distribution, and the standard deviation (the square root of the variance) indicates the spread of

the bell, with roughly 68% of the area within 1 standard deviation of the peak.

The normal distribution arises so frequently in applications due to an amazing fact: If you take a

bunch of independent random variables (with comparable variances) and average them, the result

will be roughly normally distributed, no matter what the distributions of the separate variables

might be. (This is known as the “Central Limit Theorem”.) Many interesting quantities (ranging

from IQ scores, to demand for a retail product, to lengths of shoelaces) are actually a composite

of many separate random variables, and hence are roughly normally distributed.

If X is normal, and Y = aX+b, then Y is also normal, with E[Y] = a⋅E[X] + b and

StdDev[Y] = a⋅StdDev[X]. If X and Y are normal (independent or not), then X+Y and X-Y =

X+(-Y) are also normal (intuition: the sum of two bunches is a bunch). Any normally-

distributed random variable can be transformed into a “standard” normal random variable (with

mean 0 and standard deviation 1) by subtracting off its mean and dividing by its standard

deviation. Hence, a single tabulation of the cumulative distribution for a standard normal random

variable (attached) can be used to do probabilistic calculations for any normally-distributed

random variable.

The exponential distribution : Consider the time between successive incoming calls at a

switchboard, or between successive patrons entering a store. These “interarrival” times are

typically exponentially distributed. If the mean interarrival time is 1/λ (so λ is the mean arrival

rate per unit time), then the variance will be 1/λ^2 (and the standard deviation will be 1/λ ). The

right-hand graph above displays the graph of the exponential density function when λ = 1.

Generally, if X is exponentially distributed, then Pr(s < s < X ≤ t) = e-λs^ - e-λt^ (where

e ≈ 2.71828).

The main use of the beta distribution is to “fit” it to observed data when building a model of a

real-world phenomenon: The beta distribution can take a wide variety of shapes, as seen in the

graphs below.

The lefthand graph arises when α = 5 and β = 3 , the center when α = 1.5 and β = 3, and the

right when α = 0.5 and β = 0.5. Generally, if α > 2, the density has a slope of 0 at x = 0, if

2 > α > 1, the density is near-vertical near x = 0, and if 1 > α > 0, the density rises as x

approaches 0. (Analogous properties involving β hold when x is near 1.) When α = β = 1,

the beta distribution is uniform on [0,1].

Right-Tail Probabilities of the Normal Distribution

  • +0.01 +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.
  • 0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641 0.
  • 0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247 0.
  • 0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859 0.
  • 0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483 0.
  • 0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121 0.
  • 0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776 0.
  • 0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451 0.
  • 0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148 0.
  • 0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867 0.
  • 0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611 0.
  • 1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379 0.
  • 1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170 0.
  • 1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 0.
  • 1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823 0.
  • 1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681 0.
  • 1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559 0.
  • 1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455 0.
  • 1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367 0.
  • 1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294 0.
  • 1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233 0.
  • 2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183 0.
  • 2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143 0.
  • 2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110 0.
  • 2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084 0.
  • 2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064 0.
  • 2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048 0.
  • 2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036 0.
  • 2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026 0.
  • 2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019 0.
  • 2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014 0.