Throw N balls at random into B boxes. Let a be the average number of balls, N/B,
in a box. Let P(x) be the probability that a given box has exactly x balls in it.
(a) Show that
P(x) ≈
a
x
e
−a
x!
.
Certain assumptions are needed for this expression to be valid. What are
they?
(b) Show that if a is large, the above Poisson distribution essentially becomes a
Gaussian distribution,
P(x) = a
x
e
−a
x!
≈
e
−(x−a)
2/2a
√
2πa
Throw N balls at random into B boxes. Let a be the average number of balls, N/B,
in a box. Let P(x) be the probability that a given box has exactly x balls in it.
(a) Show that
P(x) ≈
a
x
e
−a
x!
.
Certain assumptions are needed for this expression to be valid. What are
they?
(b) Show that if a is large, the above Poisson distribution essentially becomes a
Gaussian distribution,
P(x) = a
x
e
−a
x!
≈
e
−(x−a)
2/2a
√
2πa
in a box. Let P(x) be the probability that a given box has exactly x balls in it.
(a) Show that
P(x) ≈
a
x
e
−a
x!
.
Certain assumptions are needed for this expression to be valid. What are
they?
(b) Show that if a is large, the above Poisson distribution essentially becomes a
Gaussian distribution,
P(x) = a
x
e
−a
x!
≈
e
−(x−a)
2/2a
√
2πa
Swag , 11 Years ago
Grade 11
