Formulae used for tutorials in Random phenomena course - 4th set
Probability distributions of continuous random variables
Uniform distribution
A uniform random variable X over the interval [a, b] has the following probability density function
f (x) and cumulative distribution function F (x):
Z x
1
x−a
f (x) =
f (u) du =
,
F (x) =
,
for
a ≤ x ≤ b.
(1)
b−a
b−a
a
Exponential distribution
An exponential random variable X with the mean 1/λ > 0 has the following probability density
function f (x) and cumulative distribution function F (x):
Z x
−λx
f (x) = λ e
,
F (x) =
f (u) du = 1 − e−λx ,
for
x ≥ 0.
(2)
0
Normal (Gaussian) distribution
A normal random variable X with the mean m ∈ R and the standard deviation σ > 0 has the following
probability density function f (x) and cumulative distribution function F (x):
(x − m)2
2σ 2
e
,
Z x
x−m
1
F (x) =
f (u) du = 0.5 + Φ
,
f (x) = √
σ
σ 2π
−∞
Here, Φ x−m
= Φ (z) denotes the Laplace function:
σ
−
1
Φ(z) = √
2π
Z
u2
e 2 du
z −
with properties
Φ(∞) = 0.5
and
for
x ∈ R . (3)
Φ(−z) = −Φ(z) .
(4)
0
To be able to express the F (x) by the Laplace function, a standard normal random variable
Z = (X − m)/σ has been introduced.
The probability density function of a normal random variable X is usually shortly denoted by N (x; m, σ).
The probability density function of the corresponding standard normal random variable Z is always
N (z; 0, 1) and its probabilities can be calculated using the table A.1 of the textbook Opis naključnih
pojavov. The transformation N (x; m, σ) → N (z; 0, 1) is referred to as standardizing.
Approximations by normal distribution
Binomial distribution can be approximated by the normal distribution if parameter n of a binomial
distribution is large and the probability of a success is p ≈ 0.5. The parameters of a normal distribution
are then:
p
m = np
and
σ = np(1 − p) .
(5)
By an alternative criteria the approximation is good when np > 5 and n(1 − p) > 5.
Poisson distribution can be approximated by the normal distribution if λ > 5. The parameters of
a normal distribution are then:
√
m=λ
and
σ = λ.
(6)
Problems for tutorials in Random phenomena course - 4th set
Probability distributions of continuous random variables
1. The thickness of a flange on an aircraft component is uniformly distributed between 0.95 and 1.05 mm.
(a) Determine the cumulative distribution function of flange thickness. R: F (x) = 10 mm−1 (x − 0.95 mm)
(b) Determine the proportion of flanges that exceeds 1.02 mm. R: P = 0.3
(c) What thickness is exceeded by 90% of the flanges? R: x = 0.96 mm
2. The time to failure of a certain type of computer hard disk is exponentially distributed with a mean of
25 000 h.
(a) What is the probability that the disc runs without failure for at least 30 000 h? R: P = 0.301
(b) What is the time to failure that at most 10 % of discs exceed? R: t > 57565 h
3. Two weeks after being sowed, the mean plant height is 10 cm with the standard deviation of 1 cm. It is
assumed that the plant height is normally distributed.
(a) What is the probability that the height of a randomly chosen plant falls between 9 and 12 cm? R:
P = 0.819
(b) What height is exceeded by 90 % of the plants? R: h = 8.72 cm
4. The probability of getting a bad product in a series of 1000 pieces is 0.02.
(a) What is the probability that more than 30 bad products are found in a randomly selected series? R:
P = 0.0126 and P = 0.0119
(b) What is the minimum capacity of a warehouse in which all bad products from a selected series can
be stored with a probability of 0.95? R: C ≥ 28
Note: To solve some of the problems a tabulated Gaussian probability distribution is required (Table A.1 in
the textbook Opis naključnih pojavov ).