DISTRIBUTIONS e'g' MENDELs PEAS

1
DISTRIBUTIONS - e.g. MENDELs PEAS
2
P.D.F./C.D.F.

• If X is a R.V. with a finite countable set of
  possible outcomes, x1 , x2,.., then the
  discrete probability distribution of X
• and D.F. or C.D.F.
• While, similarly, for X a R.V. taking any value
  along an interval of the real number line
• So if first derivative exists, then
• is the
  continuous pdf, with

3
EXPECTATION/VARIANCE

• Clearly,
• and

4
Moments and M.G.Fs

• For a R.V. X, and any non-negative integer k, kth
  moment about the origin, defined as expected
  value of X k
• Central Moments (about Mean) 1st 0 i.e.
  EX?, second variance
• To obtain Moments, use Moment Generating Function
• If X has a p.d.f. f(x), mgf is expected value of
  e tX
• For a continuous variable, then
• For a discrete variable, then
• Generally rth moment of the R.V. is rth
  derivative evaluated at t0

5
Examples- m.g.f.s

• Suppose
• Then
• i.e.
  M.g.f. MX(t) for exponential t ltb
• Suppose p.d.f.

6
PROPERTIES - Expectation/Variance etc.
Distribution Functions

• As for R.V.s generally. For X a discrete R.V.
  with p.d.f. pX, then for any real-valued
  function g
• e.g.
  
• 
  Applies for more than 2 R.V.s also
• Variance - again has similar properties to
  previously
• e.g.
• 
  

7
MENDELs Example

• Let X record the no. of dominant A alleles in a
  randomly chosen genotype, then X a R.V. with
  sample space S 0,1,2
• Outcomes in S correspond to events
• Note Further, any function of X is also a R.V.
• Where Z is a variable for seed character
  phenotype

8
Example contd.

• So that, for Mendels data,
• and
  with
• and

9
JOINT/MARGINAL DISTRIBUTIONS

• Joint cumulative distribution of X and Y,
  marginal cumulative for X, without regard to Y
  and joint distribution (p.d.f.) of X and Y then,
  respectively
• where similarly for continuous case e.g. (2)
  becomes

10
Example Backcross 2 locus model (AaBb ? aabb)
Observed and Expected
frequencies Genotypic S.R 11 Expected S.R.
crosses 1111

• Cross
• Genotype 1 2
  3 4 Pooled
• Frequency AaBb 310(300) 36(30) 360(300)
  74(60) 780(690)
• Aabb 287(300) 23(30)
  230(300) 50(60) 590(690)
• aaBb 288(300)
  23(30) 230(300) 44(60) 585(690)
• aabb 315(300)
  38(30) 380(300) 72(60) 805(690)
• Marginal A Aa 597(600) 59(60)
  590(600) 124(120) 1370(1380)
• aa 603(600)
  61(60) 610(600) 116(120) 1390(1380)
• Marginal B Bb 598(600) 59(60)
  590(600) 118(120) 1365(1380)
• bb 602(600)
  61(60) 610(600) 122(120) 1395(1380)
• Sum 1200 120
  1200 240 2760

11
CONDITIONAL DISTRIBUTIONS

• Conditional distribution of X, given that Yy
• where for X and Y independent
  and
• Example Mendels expt. Probability that a round
  seed (Z1) is a homozygote AA i.e. (X2)

12
Standard Statistical DistributionsImportance
Modelling practical applications
Mathematical properties are known Described
by few parameters, which have natural
interpretations.Bernoulli Distribution.This is
used to model a trial which gives rise to two
outcomes success/ failure male/ female, 0 / 1.
Let p be the probability that the outcome is
one and q 1 - p that the outcome is zero.
EX p (1) (1 - p) (0)
p VARX p (1)2 (1 - p) (0)2 - EX2 p (1
- p).
Prob
1
p
1 - p
0
1
p
13
Standard distributions - Binomial
Binomial Distribution.Suppose that we are
interested in the number of successes X in n
independent repetitions of a Bernoulli trial,
where the probability of success in an
individual trial is p. Then ProbX k nCk
pk (1-p)n - k, (k 0, 1, , n) EX
n p VARX n p (1 - p)
(n4, p0.2)
Prob
1
4
np
This is the appropriate distribution to use in
modelling e.g. Number of recombinant gametes
produced by a heterozygous parent for a 2-locus
model - extension for gt3 loci is multinomial
14
Standard distributions - Poisson

• Poisson Distribution.The Poisson distribution
  arises as a limiting case of the binomial
  distribution, where n , p 0 in such a way
  that np l ( Constant)
• PX k exp ( - l ) lk / k ! (k 0,
  1, 2, ). E X lVAR X l.Poisson is
  used to model No.of occurrences of a certain
  phenomenon in a
• fixed period of time or space, e.g. O
  particles emitted by radioactive source in fixed
  direction for ? T O people arriving in a
  queue in a fixed interval of time
• O genomic mapping functions, e.g.
  crossing over as a random
• event

1
5
X
15
Standard distributions Geometric and Negative
Binomial in brief
Prob

• Geometric. This arises in the time or No.
• of steps k to the first success in a series
  of
• independent Bernoulli trials. The density
  is ProbX k p (1 - p) k -1 (k 1, 2,
  ). EX 1/p VAR X (1 - p) /p2
• Negative Binomial This is used to model the
  number of failures k that occur before the rth
  success in a series of independent Bernoulli
  trials. The density is Prob X k r k -1Ck
  p r (1 - p)k (k 0, 1, 2, )
  E X r (1 - p) / p Added Note
  Alternative form - VARX r (1
  - p) / p2 based directly on
  No. successes
• 
  - see Tables

1
X
16
Standard distributions Hypergeometric

• Consider a population of M items, of which W are
  deemed to be successes. Let X be the number of
  successes that occur in a sample of size n, drawn
  without replacement from the population. The
  density is
• Prob X k WCk M-WCn-k / MCn
  ( k 0, 1, 2, )
• Then E X n W / M VAR X n W (M -
  W) (M - n) / M2 (M - 1)
• Sampling without replacement from a finite
  population

17
Standard p.d.f.s Gamma and Exponential in brief

• The Gamma distribution e.g. from queueing theory,
  -time to the arrival of the nth customer in
  single-server queue, (mean arrival rate l).
  P.d.f. written in terms of gamma function
• or directly
• 
  with E X n / l and VAR X n / l 2
• Exponential special case of the Gamma
  distribution with n 1 used e.g. to model
  inter-arrival time of customers, or time to
  arrival of first customer, in a simple queue,
  fragment lengths in genome mapping etc.
• The p.d.f. is f (x) l exp ( - l x
  ), x ³ 0, l gt 0 0 otherwise
  

18
Standard p.d.f.s - Gaussian/ Normal

• A random variable X has a normal distribution
  with mean m and standard deviation s if it has
  density
• and
• Arises naturally as the limiting distribution of
  the average of a set of independent, identically
  distributed random variables with finite
  variances.
• Plays a central role in sampling theory and is a
  good approximation to a large class of empirical
  distributions. Default assumption ?in many
  empirical studies is that each observation is
  approx. Normally.
• Statistical tables of the Normal distribution
  are of great importance in analysing practical
  data sets. X is said to be a Standardised Normal
  variable if m 0 and s 1.

19
Standard p.d.f.s Students t-distribution

• A random variable X has a t -distribution with n
  d.o.f. ( tn ) if it has density
  
  0 otherwise.Symmetri
  cal about the origin, with EX 0 VAR X
  n / (n -2).
• For small n, the tn distribution is very flat.
  For n ³ 25, the tn distribution ? standard normal
  curve.
• Suppose Z a standard Normal variable, W has a
  cn2 distribution and Z and W independent then
  r.v.
• If x1, x2, ,xn is a random sample from N(m ,
  s2) , and, if define
  
• 
  then

20
Chi-Square Distribution

• A r.v. X has a Chi-square distribution with n
  degrees of freedom (n a positive integer) if it
  is a Gamma distribution with l 1, so its p.d.f.
  is
• EX n Var X 2n
• Two important applications- If X1, X2, , Xn
  a sequence of independently distributed
  Standardised Normal Random Variables, then the
  sum of squares
• X12 X22 Xn2 has a ?2 distribution
  (n degrees of freedom).
• - If x1, x2, , xn is a random sample from
  N(m ,s2), then
• and
  and
• s2 has ?2 distribution, n - 1 d.o.f., with r.v.s
  and s2 independent.

Prob
c2 n (x)
X
21
F-Distribution

• A r.v. X has an F distribution with m and n
  d.o.f. if it has a density function ratio of
  gamma functions for xgt0 and 0 otherwise.
• 
  and
• 
  
• For X andY independent r.v.s, X cm2 and Y
  cn2 then
• One consequence if x1, x2, , xm ( m ³ 2) is
  a random sample from N(m1, s12), and y1, y2, ,
  yn ( n ³ 2) a random sample from N(m2,s22),
  then