1.Continuous Univariate Distributions:
Continuous sample space: Definition, Illustrations, Continuous random variable: Definition, Probability density function (p.d.f.), Cumulative distribution function (c.d.f.), Properties of c.d.f. (without proof), Probabilities of events related to random variable.
Expectation of continuous r.v., Expectation of function of r.v. E [g(x)], Mean, Variance, Geometric mean, Harmonic mean, Raw and central moments, Skewness, Kurtosis. Mean deviation about mean.
Moment generating function (M.G.F.): Definition and properties, Cumulant generating function (C.G.F.): Definition, Properties. 1.4 Mode, Partition values (Q1, Q2, Q3) Deciles. Percentiles.
Probability distribution of function of r.v. Y = g(X) using (i) Jacobian of transformation for g() monotonic function and one-to-one, on to functions, (ii) Distribution function for Y = x2, Y = X etc., (iii) M.G.F. of g(X).
2.Continuous Bivariate Distributions:
Continuous bivariate random vector of variable (X, Y): Joint p.d.f., Joint c.d.f., Properties (without proof), Probabilities of events related to r.v. (events in terms of regions bounded by regular curves, Circles, Straight lines). Marginal and conditional distributions.
Expectation (r.v.), Expectation of function of r.v. E [g(X, Y)], Joint moment, Cov (X, Y), Corr (X, Y), Conditional mean, Conditional variance, E [EXY = y)] = E(X), Regression as a conditional expectation. Theorems on expectation:
(i) E (X + Y) = E (X) + E (Y), (ii) E (XY) = E (X) E (Y), if X and Y are independent, Generalization to k variables E (ax + bY + c), Var (ax + by + c). (Statement only proof not expected).
Independence of r.v. (X, Y) and its extension of k dimensional r.v.
M.G.F.: Mx, y (t1, t2), Properties, M.G.F. of marginal distribution of r.v.s., Properties.
(i) Mx, y (t1, t2)= Mx (t1, 0) My (0, t2), if X and Y are independent r.v.s.
(ii) Mx+y (t) = Mx, y(t, t).
(iii) Mx + y (t) = Mx (t) My (t) if X and Y are independent r.v.s.
Probability distribution of transformation of bivariate r.v.
U = 1 (X, Y), V = 92 (X, Y).
3.Standard Univariate Continuous Distributions:
Uniform or Rectangular Distribution
Probability density function (p.d.f.) f(x) = 3
Notation: X ---> U [a, b].
(03 L)
b a
;
a≤x≤b
0
; otherwise
p.d.f., Sketch of p.d.f., c.d.f., Mean, Variance, Symmetry,
Distribution of (i)
X-a b-X (ii) b-a' b-a
(iii) Y = F(X), where F(X) is the
c.d.f. of continuous r.v. X.
Application of the result of model sampling. (Distributions of
X + Y, X-Y, XY and X/Y are not expected.)
Probability density function (p.d.f.).
f(x) = σ2π
0
Notation: X N (u, o2).
(10 L)
; - ∞ < x < ∞ ∞ <μ < ∞; σ > 0
;
otherwise
p.d.f. curve, Identification of scale and location parameters, Nature of probability curve, Mean, Variance, M.G.F., C.G.F., Central moments, Skewness, Median, Mode, Quartiles (Q1, Q2, Q3), Points of inflexion of probability curve normal probability integral tables, Mean deviation, Additive property, Probability distribution X-μ of: (i) Standard normal variable (S.N.V.), (ii) aX+ b, (iii) ax + by + c, where X and Y are independent normal variates. Probability distribution of X, the mean of n i.i.d. N (u, o2) computation of probability.
σ
Central limit theorem (CLT) for i.i.d. r.vs with finite positive variance. (Statement only). Its illustration for Poisson and Binomial distributions.
Probability density function (p.d,f.).
(04 L)
ae
; x≥ 0; α > 0
f(x) =
0
;
ot
I Nature of Density curve, Interpretation of a as a interarrival rate 1 α
of customer joining the queue and as mean, Mean, Variance,
M.G.F., C.G.F., skewness, Kurtosis, c.d.f., Graph of c.d.f., Lack of memory property, Quartiles (Q1 Q2 Q3). Mean devidation about mean, distribution of sum of two i.i.d. exponential r.v.s. Distribution of Min (X, Y) and Max (X, Y) with X, Y i.i.d.
exponential r.v.s.
Γλ
f(x) = xλ-1 e-xx
;
;
x ≥ 0, α, λ > 0 otherwise
Notation: XG (a, 2), Nature of probability curve, special cases: (i) α = 1, (ii) λ = 1, M.G.F., C.G.F., moments, cumulants, skewness, kurtosis, mode, additive property. Distribution of sum of n i.i.d. exponential variables. Relation between distribution function of Poisson and Gamma variates.