Question

If X∼Binom(n,p), E[X] = np. Calculate V[X] = np(1−p) by:

a) First show E[X(X−1)] + E[X] − (E[X])^2 = V[X] (Hint: Use propertie of E[·] and V[·]).

b) Show E[X(X−1)] = n(n−1)p^2

c) Use E[X] = np, a) and b) to discuss, V[X] = np(1−p).

Answer #1

a)

Now,

b)

c) Variance of X =

suppose that X ~ Bin(n, p)
a. show that E(X^k)=npE((Y+1)^(k-1)) where Y ~ Bin(n-1, p)
b. use part (a) to find E(x^2)

The normal approximation of the binomial distribution is
appropriate when:
A. np 10
B. n(1–p) 10
C. np ≤ 10
D. np(1–p) ≤ 10
E. np 10 and n(1–p) 10

Suppose X is binomial random variable with n = 18 and p = 0.5.
Since np ≥ 5 and n(1−p) ≥ 5, please use binomial distribution to
find the exact probabilities and their normal approximations. In
case you don’t remember the formula, for a binomial random variable
X ∼ Binomial(n, p), P(X = x) = n! x!(n−x)!p x (1 − p) n−x . (a) P(X
= 14). (b) P(X ≥ 1).

Show that if X ∈ N(µ, σ2 ), then E(X) = µ, and V ar(X) = σ 2

Suppose that x has a binomial distribution with n
= 202 and p = 0.47. (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) (Click to select)≥≤
5
(b)...

Suppose that x has a binomial distribution with n = 199 and p =
0.47. (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) (Click to select) 5 (b)...

For a binomial random variable X, if np and n(1-p) are both at
least 5 what can we do?

Let G = (V,E) be a graph with n vertices and e edges. Show that
the following statements are equivalent:
1. G is a tree
2. G is connected and n = e + 1
3. G has no cycles and n = e + 1
4. If u and v are vertices in G, then there exists a unique path
connecting u and v.

The normal approximation of the binomial distribution is
appropriate when
np ≥ 5.
n(1 − p) ≥ 5.
np ≤ 5.
n(1 −
p) ≤ 5 and np ≤ 5.
np ≥ 5 and n(1 − p) ≥ 5.

Compute E(N^2), for N∼Geom(p), by conditioning on the outcome of
the first trial. Use this to calculate Var(N)

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