BIO NOMINAL DISTRIBUTION

• BIO NOMINAL DISTRIBUTION
  RESEARCH METHODOLOGY
  
  
• MEANING

1. IS DISCRETE PROBABILITY DISTRIBUTION
2. DISCOVERED BY JAMES BERNOULLI
3. USED IN SUCH SITUATIONS WHERE AN EXPERIMENT RESULTS IN TWO POSSIBILITIES SUCCESS OR FAILURE
4. IS A DISCRETE PROBABILITY DISTRIBUTION WHICH EXPRESSES THE PROBABILITY OF ONE SET OF TWO ALTERNATIVES-SUCCESS(p) AND FAILURE(q)

• DEFINITION OF BIO NOMINAL DISTRIBUTION

1. BIO NOMINAL DISTRIBUTION IS DEFINED AND GIVEN BY THE FOLLOWING PROBABILITY FUNCTION
2. P(X=x)=ⁿC_xq^n-x.p^x
3. p=PROBABILITY OF SUCCESS
4. q=PROBABILITY OF FAILURE=1-p.
5.  n=NUMBER OF TRIALS
6. P(X=x)=PROBABILITY OF x SUCCESS IN n TRIALS

• ASSUMPTION

1. NUMBER OF TRIALS IS FINITE
2. EACH TRIAL MUST RESULT IN TWO MUTUALLY EXCLUSIVE OUTCOME SUCCESS OR FAILURE
3. IN EACH TRIAL,THE PROBABILITY OF SUCCESS DENOTED BY P REMAINS CONSTANT. LIKE IF WE TOSS A COIN,THE PROBABILITY OF GETTING A HEAD IN EACH TOSS REMAINS THE SAME=1/2
4. TRIALS ARE INDEPENDENT: THE OUTCOME OF ANY TRIAL DOES NOT EFFECT THE OUTCOME OF SUBSEQUENT TRIALS

• PROPERTIES OF BIONOMINAL DISTRIBUTION

1. THE BIO NOMINAL DISTRIBUTION IS A THEORETICAL FREQUENCY DISTRIBUTION WHICH IS BASED ON BIO NOMINAL ALGEBRA
2. IS DISCRETE PROBABILITY DISTRIBUTION IN WHICH THE NUMBER OF SUCCESS 0,1,2,3----n ARE GIVEN IN WHOLE NUMBER NOT IN FRACTION NUMBER
3. CAN BE PRESENTED GRAPHICALLY BY MEANS OF A LINE GRAPH. THE UMBER OF SUCCESS IS TAKEN ON X-AXIS AND THE PROBABILITY OF SUCCESS TAKEN ON THE Y AXIS
4. THE SHAPE OF BIONOMINAL DISTRIBUTION DEPENDS ON VALUE OF p and q

• MAIN PARAMETERS

1. MEAN =X‾ =np

a.    VARIANCE=² =npq

b.    S.D==   =√npq

2. MOMENT COEFFICIENT OF SKEWNESS=√β1 =q-p/(√npq)

3. MOMENT OF COEFFICIENT OF KURTOSIS=β2 =3 +(1-6pq)/√npq

• APPLICATION OF BIONOMINAL DISTRIBUTION

1. APPLICATION OF BIO NOMINAL DISTRIBUTION:- WHEN WE GIVEN THE PROBABILITY OF OCCURRENCE OF AN EVENT RELATING TO PROBLEM IE THE VALUE OF p and q ARE GIVEN WE CAN FIND THE PROBABILITY OF THE HAPPENING OF EVENT EXACTLY X TIMES OUT OF N TRIALS BY USING THE FORMULA   P(X=x) = ⁿC_xq^n-x.p^x
2. TO FIND n,p and q FROM MEAN AND STANDARD DEVIATION
3. TO FIND MEAN AND STANDARD DEVIATION WHEN n,p and q ARE GIVEN
4. FITTING OF BIO NOMINAL DISTRIBUTION

• PROBLEM:-
•  A FAIR COIN IS TOSSED THRICE. FIND THE PROBABILITY OF GETTING
• EXACTLY TWO HEADS
• AT LEAST TWO HEADS
• AT THE MOST 2 HEADS  

• SOLUTION
• LET p= PROBABILITY OF GETTING ONE HEAED AND WHEN A COIN IS TOSSED =1/2
•  q= THE PROBABILITY OF TAIL=1/2
•  n = 3
• P(X=x) =ⁿC_xq^n-x.p^x
• P (2H)=³C₂q^3-2.p²  =³C₂ (1/2)¹.(1/2)² =3X1/2X ¼=3/8

• P =AT LEAST 2 HEADS
• =P(2H)+P(3H)
• P(2H) =P(X=x) =ⁿC_xq^n-x.p^x
• P (2H)=³C₂q^3-2.p²  =³C₂ (1/2)¹.(1/2)² =3X1/2X ¼=3/8
• P(3H)=P(X=x) =ⁿC_xq^n-x.p^x
• P (3H)=³C₃q^3-3.p³  =³C₃ (1/2)⁰.(1/2)³ =1X1/8
• P(2H)+P(3H) = 3/8 +1/8=4/8=1/2

• AT THE MOST TWO HEADS
• P (AT TWO MOST TWO HEADS)
• = P(0H)+P(1H)+P(2H)
• 1-P(3H)
• P (3H)=³C₃q^3-3.p³  =³C₃ (1/2)⁰.(1/2)³ =1X1/8
• AT MOST TWO HEAD=1-1/8=7/8

• TO FIND n.pand q FROM MEAN AND STANDARD DEVIATION

1. THE MEAN OF A BIO NOMINAL DISTRIBUTION IS 20 AND SD IS 4. FIND n.p and q
2. MEAN= np
3. S.D=√npq

4.     MEAN =np=20 -------1

5.     = √ npq =4

6.     SQUARING BOTH SIDES

7.     npq= 16 -----2

8.     DIVIDING 2 BY 1

9.     npq/np=16/20=4/5

10.  p=1-4/5=1/5

11.  np=20=nx1/5=20=n=20x5=100