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JRANDOM W ARIABLES, , , , , , , , , , IMPORTANT DEFINITIONS, AND RESULTS, , , , , , , , , , E DISCRETE RANDOM VARIABLES, , ‘J, Define discreterandom variables:, , A discrete random variable is a R.V. X whose possible, _ values constitute finite set of values countably infinite set of, values., , 2. Define distribution function of the r.v., The: distribution function of a random variable xX, defined in (—0, 0) is givenby, F(x) = P(X<x)=P {s:X(s)sx}, 3. Write any 4 properties of distribution functions. ; :, Property 1: P(a<X<b)= F(b)-F(@), i : where F(x) = P(X sx), Property 2: P(a<X<b) = P(X=a)+F(b)- F(a), Property3.: P(a<X<b) = P(a<X<b)-P(K=b), 2 Property 4: P(a<X<b) =: P(a<X<b)+P(X=a), 4, Important Notes On Distribution Function F@:, , Let the random variable X take values x1, X9),--++++ o®p, with probabilities py, 2, ..+5 Py and let xy < x2 3S. <X,), Then we have, F (x)= P(X <x1) =0,.-0 <x <2}, , F (x) =P (X <x) = P(X <x) +P (X= ay) = OF P= Pr, Es P(X $x) =P(X <x) +P (X= ay) + P (X= 42), , , , , , , , , , , , =PitP2, F (x) = P(X <x,) = POX <4) + PK = x) +. + P(X = Xp), =p, tpot aon +p, = 1, , Scanned by CamScanner

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1.2 VICTORY Suing, , 5. Define probability, , Let X be a one, , the values X45 X25 ¥3y ++, , can associate a number Pj 12, ‘called the probability of x,, , The numbers p; = P (x;) satisfies the following conditions, , mass function (or) probability Sunction,, dimensional discrete R.V. which tak,, To each possible outcome ‘x,, me, , PK =x) = P(t) =,, , () Pm) 20, Vi (ii) BP eal, | continuous RANDOM VARIABLES _|, , 6. Define continuous random variables., A RV. ‘X” which takes all possible values in a given, interval is called'a Continuous Random Variable., 7, Define probability density function., If there is a function y= f (x) such that | ~, , P@<X<x+Ayn", ses oa ane Oi, , then this function f (x) is termed as: the probability density, function (or) simply density function of the r.v..‘X’. It is also, , called the frequency function, Tiago density or the, probability Benes), , , , , , b ‘, 8. I$) pa, then Packs). = S$) ax, , a, , , , 9. Write the paneled of p.d, if, The pdf. f (x) of a R.V. X has. the following, , properties., , (i) fG)20,-o<x<0 (ii) Sf@dr=1, , 10. (i) P(X=Q=0, (ii) Pia =X<b)= P(a <X<b), =P(@<X <b)= P(a<X<b), , , , , , (errs —_, Scanned by CamScanner

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RANDOM VARIABLES “3, , 11_Important formuta in interms of p.df., , (i) | Arithmetic mean, , , , , , , , , , , , , , xf (x) dx, , ii Harmoni 1, (ii) ic mean 5 L(x) dx, , .. | Geometric mean ‘G’, (iii) ogG log xf (x) dx, , ; Moments about, (iv) origin b,! x" f(x)d, , Moment about any. () “point A Hy! (e-Ay'f @) ae, , , , Moment about mean ., , , , , , the mean is" M.D., , 12. Define distribution function f (x). , If f (xis a p.d.f. of a continuous random variable ‘X’,, then the function, , , , , , , , aoe lacoelacelsacclacoclacie[acjclacs, , , , _ vi) ik (x - mean)’ f (x) dx, , r, (vii) | Variance By (x - mean)? f (x) dx, (viii) Mean deviation about |x mean |f (x) dx ;, , , , Fy.(x) =F @) = P(X sx) =. { f(x)dx,-a<x<o, , ; —e, is called- the distribution: function or cumulative, distribution function of the random variable ‘X’., , _ 13. Write any 4 properties of c.d.f., , (i) O<F@)s1, -2<x<0, (ii), Mt F@) = 0, , Fee, (iii) P(asX<b) = F(b)-F(a), d F(x), (iv) F(@) = a =f), , (or) dF (x) = fQ@)dx, , , , Scanned by CamScanner