Question
In: Statistics and Probability
1. Suppose X has a binomial distribution with parameters n and p. Then its moment-generating function...
1. Suppose X has a binomial distribution with parameters n and p. Then its moment-generating function is M(t) = (1 − p + pe^t ) n .
(a) Use the m.g.f. to show that E(X) = np and Var(X) = np(1 − p).
(b) Prove that the formula for the m.g.f. given above is correct. Hint: the binomial theorem says that Xn x=0 n x a^x b^(n−x) = (a + b)^n .
2. Suppose X has a Poisson distribution with parameter λ. Then its moment-generating function is M(t) = e^(λ(e t−1)) .
(a) Use the m.g.f. to show that E(X) = λ and Var(X) = λ.
(b) Prove that the formula for the m.g.f. given above is correct. Hint: for any real number a, a^x / x! = e a .
Solutions
orchestra answered 2 years agoRelated Solutions
Binomial Distribution. Suppose that X has a binomial distribution with n = 50 and p =...
Binomial Distribution. Suppose that X has a binomial
distribution with n = 50 and p = 0.6. Use Minitab to simulate 40
values of X.
MTB > random 40 c1;
SUBC > binomial 50 0.6.
Note: To find P(X < k) for any k > 0, use ‘cdf’ command;
this works by typing:
MTB > cdf;
SUBC > binomial 50 0.6.
(a) What proportion of your values are less than 30? (b) What is
the exact probability that X will...
The p.d.f of the binomial distribution random variable X with parameters n and p is f(x)...
The p.d.f of the binomial distribution random variable X with
parameters
n and p is
f(x) =
n
x
p
x
(1 − p)
n−x x = 0, 1, 2, ..., n
0 Otherwise
Show that
a) Pn
x=0 f(x) = 1 [10 Marks]
b) the MGF of X is given by [(1 − p) + pet
]
n
. Hence or otherwise show
that E[X]=np and var(X)=np(1-p).
Suppose that x has a binomial distribution with n = 201 and p = 0.49. (Round...
Suppose that x has a binomial distribution with
n = 201 and p = 0.49. (Round np and
n(1-p) answers to 2 decimal places. Round your answers to 4 decimal
places. Round z values to 2 decimal places. Round the intermediate
value (σ) to 4 decimal places.)
(a) Show that the normal approximation to the
binomial can appropriately be used to calculate probabilities about
x.
np
n(1 – p)
Both np and n(1 – p) ≤
or ≥ 5
(b)...
Suppose that x has a binomial distribution with n = 50 and p = 0.6, so...
Suppose that x has a binomial distribution with n = 50 and p =
0.6, so that μ = np = 30 and σ = np(1 − p) = 3.4641. Approximate
the following probabilities using the normal approximation with the
continuity correction. (Hint: 25 < x < 39 is the same as 26 ≤
x ≤ 38. Round your answers to four decimal places.)
(a) P(x = 30)
(b) P(x = 25) (
c) P(x ≤ 25)
(d) P(25 ≤...
Let X have a binomial distribution with parameters n = 25 and p. Calculate each of...
Let X have a binomial distribution with parameters
n = 25
and p. Calculate each of the following probabilities
using the normal approximation (with the continuity correction) for
the cases
p = 0.5, 0.6, and 0.8
and compare to the exact binomial probabilities calculated
directly from the formula for
b(x; n, p).
(Round your answers to four decimal places.)
(a)
P(15 ≤ X ≤ 20)
p
P(15 ≤ X ≤ 20)
P(14.5 ≤ Normal ≤ 20.5)
0.5
1
2
0.6...
Let X have a binomial distribution with parameters n = 25 and p. Calculate each of...
Let X have a binomial distribution with parameters
n = 25
and p. Calculate each of the following probabilities
using the normal approximation (with the continuity correction) for
the cases
p = 0.5, 0.6, and 0.8
and compare to the exact binomial probabilities calculated
directly from the formula for
b(x; n, p).
(Round your answers to four decimal places.)
P(20 ≤ X)
p
P(20 ≤ X)
P(19.5 ≤ Normal)
0.5
0.6
0.8
Let X have a binomial distribution with parameters n = 25 and p. Calculate each of...
Let X have a binomial distribution with parameters
n = 25
and p. Calculate each of the following probabilities
using the normal approximation (with the continuity correction) for
the cases
p = 0.5, 0.6, and 0.8
and compare to the exact binomial probabilities calculated
directly from the formula for
b(x; n, p).
(Round your answers to four decimal places.)
(a)
P(15 ≤ X ≤ 20)
p
P(15 ≤ X ≤ 20)
P(14.5 ≤ Normal ≤ 20.5)
0.5
0.6
0.8
(B)P(X...
Suppose the number of cars in a household has a binomial distribution with parameters n =...
Suppose the number of cars in a household has a binomial
distribution with parameters n = 20, and p = 55 %. Find the
probability of a household having
(a) 10 or 13 cars
(b) 11 or fewer cars
(c) 16 or more cars
(d) fewer than 13 cars
(e) more than 11 cars
Suppose the number of children in a household has a binomial distribution with parameters n =...
Suppose the number of children in a household has a
binomial distribution
with parameters n = 16, and p = 80 %.
Find the probability of a household having:
(a) 6 or 15 children
(b) 13 or fewer children
(c) 10 or more children
(d) fewer than 15 children
(e) more than 13 children
Suppose the number of cars in a household has a binomial distribution with parameters n =...
Suppose the number of cars in a household has a binomial
distribution
with parameters n = 9, and p = 25 %.
Find the probability of a household having:
(a) 0 or 2 cars
(b) 0 or fewer cars
(c) 4 or more cars
(d) fewer than 2 cars
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