SHow full worjk please The binomial random variable x has pa

SHow full worjk [please

The binomial random variable x has parameters n = 10 and p = 0.3. Find (1) P(x = 2) = b(10, 0.3, 2) P(x s 2) = B(10, 0.3, 2) P(x 2) = 1 - P(x 1) mu and Consider the binomial experiment with 15 trials and probability of success on each trial 0.40. Find (1) probability of exactly four successes. probability of more than five successes. probability of less than three successes. mean number of successes in 15 trials. A multiple choice test has 25 questions with five choices per question. Assume each question is answered by guessing. Find (1)

Solution

Binomial Distribution

PMF of B.D is = f ( k ) = ( n k ) p^k * ( 1- p) ^ n-k
Where   
k = number of successes in trials
n = is the number of independent trials
p = probability of success on each trial

a)
P( X = 2 ) = ( 10 2 ) * ( 0.3^2) * ( 1 - 0.3 )^8
= 0.2335
b)
P( X < = 2) = P(X=2) + P(X=1) + P(X=0)   
= ( 10 2 ) * 0.3^2 * ( 1- 0.3 ) ^8 + ( 10 1 ) * 0.3^1 * ( 1- 0.3 ) ^9 + ( 10 0 ) * 0.3^0 * ( 1- 0.3 ) ^10   
= 0.3828
c)

P( X < 2) = P(X=1) + P(X=0)
= ( 10 1 ) * 0.3^1 * ( 1- 0.3 ) ^9 + ( 10 0 ) * 0.3^0 * ( 1- 0.3 ) ^10   
= 0.1493
P( X > = 2 ) = 1 - P( X < 2) = 0.8507
d)
Mean ( np ) =3
Standard Deviation ( npq )= 10*0.3*0.7 = 1.4491

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