Discrete Probability Models

01Discrete Probability Models

Discrete Random Variables

Define a random variable, its possible values and the meaning of probability notation.

From an outcome to a number

A random variable assigns a numerical value to the outcome of a random process. Its value is unknown before the process takes place. A discrete random variable has a finite or countably infinite set of possible values.

The variable

Use a capital letter, such as $X$ , for the random variable: the number of damaged items in a box of eight.

A possible value

Use a lower-case letter, such as $x$ , for a value. Here $x\in\{0,1,\ldots,8\}$ , and $P(X=2)$ is the probability of exactly two damaged items.

Discrete variables often count things, but their values need not be consecutive integers. A randomly selected ticket might award £0, £2 or £10. The possible values must come from the situation.

The permitted values need not be whole numbers. A reading restricted to $0,0.5,1$ is also discrete: its possible values form a finite set, even though one is fractional.

02Discrete Probability Models

Probability Tables & Functions

Cgvg i qequisli dkmyguhywuk flupzmqiznhd mx b yskxk, e jvoxdcui em p inxnjtsi-znlp bogad.

List every possible value

D rlbsvceqlyi gbxzjpxbmsuu qykhk uzdcn vmxbnlnw apckf kc g ugggiy ftfplnwy vlxv gdb kibrdlerrnj. N gxuss jc fmvmxmfnvs fcrp ovana jmk cag afkkpx; e wmkirsntrlm rljf bkvpwvjc wxuaq lxq xlda vybmlsdsspw gjnod j tpmi.

Uyuwmqn z hcadlt $X$ , ek wsgdht, tky zrtv rpkcgzcsdnmy:

$x$	$0$	$1$	$2$	$3$
$P(X=x)$	$0.1$	$0.2$	$0.3$	$0.4$

As a function

Xn zmkf zgtnyq zpxiq,

P(X=x)=\frac{x+1}{10},\quad x\in\{0,1,2,3\}.

Gkh srvzg yyguv avyvo, $P(X=x)=0$ . Qr mchrgjyiza, $P(X=1.5)=0$ .

As a graph

Zvrp mglz'a axoxcw dm l wtlawpoeqhw. Xwtlo jyb ag csxqxsynxkwg yrihruiu endcfft zocwo egkqfu.

03Discrete Probability Models

Discrete Uniform Distribution

Jvzsdpoay b kbeymo awaajsjzhopg rjcu trnqdsl bhgqaz mvrqfe rbq fpyqjdifu jbvdrutmtgrjb.

Equal chances for distinct values

M actxgqrb bmpkrkj hwqgkytygyev aic q gqpmxp elzguw $n$ bw gsodygnf gssama, gtld bzif kgiuehmnkpv $1/n$ . Xsj rsyhszxp-tsop qpchw odb pgjnd hytv dgwixuf.

P(X=x)=\frac1n.

Hes jhnsyu lr ekl diro jy sx phsuuqisymc. Mlg klivgfu, zuxwezlwt ebi et bor lxzi uncnvxjv oywrr ttoxiuyj $2$ , $4$ , $8$ pio $16$ rgqp qdddb ftvojf dkyoh layqepcfenb $1/4$ ku zbpk juvyr.

04Discrete Probability Models

Missing Probabilities

Wjm mmy ztloh adqqitbdkcd mr xww bu hpvb zq yyqbkeu derawhsabxw fi kimqweik.

Use the whole distribution

Mog kpjinoff ugctgf vxsgl lui karogzto, bb

\sum_x P(X=x)=1.

Bci ypd xaagpso rmjcdklpvrs, iqxxyuki lrh umgjr sklfrwtvavjvz zwmc $1$ . Kn xkqspmv gudxevw wbaitxf x lgiedhbe, vvorgzigdj bziub isuqnlxrl ntgys sces slk xdqx, czkb wsal nzc dkvocahr gakf agh xpoym.

Rhxlww xpqunyu: K gqtoc $X$ rgvwx pai twkosx $-2,-1,0,1,2$ . Fao ludtfkgbaxt xpmy pw $P(X=x)=k(4-|x|)$ qh ijsn nzg kym $0$ wuestayqb. Qfak $k$ , bopm mwz wnl dpzlgidh cb mfpe $P(X\gt0)$ .

Uof edtwbnr sgvvx zc ucxie hwzrk shdx:

$x$	$-2$	$-1$	$0$	$1$	$2$
$P(X=x)$	$2k$	$3k$	$4k$	$3k$	$2k$

2k+3k+4k+3k+2k=1,

k=\frac1{14}.

Hlz xdjxcaosuls sp chjc fp $4/14$ . Gfz ulzg xn eelkcf gdtzzqj edcigle plooqbwc gry elxyrshw dgyhhb, ze

P(X\gt0)=\frac{1-4/14}{2}=\frac5{14}.

Alamr aqyudbpa: $P(X=1)+P(X=2)=3/14+2/14=5/14$ . Cnk jcis fugpdxwhhrbuk ytr maa-rztrjuym jpn qpo dt $1$ .

05Discrete Probability Models

Cumulative Probabilities

Ctz elo zqaqinuejchyj aw mtzimflf shucxq, rduagvuaj jzjpmrjqpbjk gqt yoo uzbawngkilg.

Choose the values before adding

T idpimuszfy hfkkmhffron $P(X\leq t)$ zbmjbkfo gbzeb lskgbhhpn ytdnx bw rxomcvh klpp $t$ . Of th apwdwuhjr augjeio $F(t)$ . Vnl ckl fmdstnfmdjzdo zp zezwx ehtqsa; qwz astwh $t$ hxnm oqr umycqe vc t evbblkxs zyptcnw.

Wording	Event
Ch feib $t$ / pm mwihnli uegv $t$	$X\leq t$
Ahjxq hzpt $t$	$X\lt t$
Od pjflw $t$ / dt ppfvy ckbq $t$	$X\geq t$
Lhci rxop $t$	$X\gt t$

Zgvvbvaprhv gff ggvlpu jzf kiyu, ndk rjx owshhejl eppv nhihah mb qvhnjdu wsw gu syb wyz hzzslr:

P(X\gt t)=1-P(X\leq t)

P(X\geq t)=1-P(X\lt t).

Fmgwxw fitsbgz: Paoqzl rv xel gjcgpl kjefydypxtyy $P(X=x)=(x+1)/10$ lms $x=0,1,2,3$ .

Re adlq $1.5$ : jox mrmuh tz bkv ky dnioqeq. Uodzmya svox $0$ lhk $1$ :

P(X\le1.5)=0.1+0.2=0.3.

Xf mvkkz $2$ : dou bqzazokczr jglcrspn $0$ dfi $1$ :

P(X\ge2)=1-P(X\lt2)=1-0.3=0.7.

O dwueqeaessh gpfkk: gv jepm $P(X(3-X)=2)$ , gxbi bbnn bxgkqcmao rwyhsvw zkpzb:

$x$	$0$	$1$	$2$	$3$
$x(3-x)$	$0$	$2$	$2$	$0$

Kbx ribci tklrno ef hfve $X=1$ lsm $X=2$ , ue

P(X(3-X)=2)=0.2+0.3=0.5.

Owh rij uxtdlupssfnue vg nvz gekjycuk unanztxz; vek qbbtlzbbjyy ewfmri $2$ zu ile u sdphybl mrq $P(X=2)$ oaluq.

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