Probability and Statistics for Engineers Textbook

Probability and Statistics for Engineers
Level: 1st Year Students
Section(s): All groups
Major: Mechanical/Civil Engineering

Stat & Prob. for Engineers_ENSIT2025/2026 Ali TRABELSI, Dr., Eng., 2
Chapter. 3.
RANDOM VARIABLES AND
PROBABILITY DISTRIBUTION
A. DEFINITION
 A Random Variable (R.V) is a function that associates a real number with each element in a
sample space. A R.V is either DISCRETE (e.g., nbr. of defects in a production line, nbr. of
admitted female students in medical school, etc.) or CONTINUOUS (e.g., Dia. of shafts produced
in a turning center, stress value in a part submitted to varying torque, normal/plane anisotropy of
sheet metal when subjected to a forming process, etc)
Q3.1: Complete by continuous or discrete. A R.V, which measures/counts,
1) The outlet pressure of a hydraulic pump is ……………………..…………………….…………..
2) The nbr. of accidents per year in a forging workshop is…………………………………………
3) The national height of newborn babies is ……………………………..…………………………..
4) The IQ score of the Tunisian primary students is ……….………………………………………..
Q3.2: An urn contains 4 red balls and 3 black. We draw two balls without replacement.
Suppose X is an R.V., which represents the nbr. of red balls drawn, describe X.
B. CONTINUOUS AND DISCRETE PROB. DIST
 X is a discrete R.V. The set of pairs (x, f(x)=p(X=x)) is the PROBABILITY FUNCTION or PROB.
DENSITY FUNCTION of X 
For continuous R.V, X has zero prob. of assuming exactly any of its values, hence, its prob. dist.
cannot be given in a tabular form, and intervals of continuous R.V are considered.

Stat & Prob. for Engineers_ENSIT2025/2026 Ali TRABELSI, Dr., Eng., 3
Q3.3: A shipment of 8 PCs contains 3 defective PCs. A school makes a random purchase of 2
PCs. Give, Q1. The prob. dist of the number of defective PCs. Q2. The cumulative prob. for
X=1.
Q3.4: In a fan assembling line, the proportion of defective products can be described using a
continuous R.V. Let it be X, 
Q1. Is f(x) a probability density function?
Q2. Calculate Prob. (¼<X<½).
Q3. Calculate F(⅔).
Q3.5: X is the error measurement of a physical quantity given by the density function
Q1. Determine k so that f(x) is a valid density function.
Q2. Calculate Prob. (X≤0.5) and Prob. (X ≥ 0.8).
Q3. What is Prob. (0.85≤X≤0.93)?
C. JOINT PROB. DIST.(J.P.D)
 In some situations, we need to record the outcomes of numerous RVs. SIMULTANEOUSLY,
e.g., the hardening capacity, Hc, the tensile stress (Ts), and the Yield stress (Ys) occur in a
given material subjected to forming force. This results in a 3D sample space (Hc, Ts, Ys) and
the joint prob. dist. f (Hc, Ts, Ys) is required.
Given two RVs, X and Y.
Discrete RV
Continuous RV 
Note: The J.D.P, f(x,y), is a surface lying in region A above the (x, y) plane and 
Q3.6: Two ATC systems are randomly selected from a company that commercializes three brand
names of ATCs (3ATCs of brand A, 2 ATCs of brand B, and 3 ATCs of brand C). Given X is the
nbr. of ATCs of brand A and Y of brand B. X and Y are two R.Vs

Stat & Prob. for Engineers_ENSIT2025/2026 Ali TRABELSI, Dr., Eng., 4
Q1. Find the J.P.F, f(X=x, Y=y).
Q2. Prob. {(X, Y) ϵ A, 2x-y ≥ 1}.
Q3. Prob. {(X, Y) ϵ D, x2+ (y-1)2 ≥ ¼}.
Q3.7: Given
Q1. Prove that 
Hint:  
Q2. Give Prob. (X, YϵA, 0 ≤ x≤ ½ and ¼≤ y≤ ½).
Q3. Give Prob. (X, Y ϵ A such that).
Hint:  
.5.5.5
D. MARGINAL DISTRIBUTION
 Given X and Y, two RVs. The MARGINAL DISTRIBUTION functions g(x) and h(y) are
obtained by summing/integrating f(x, y) over the values of Y and X, respectively.
Discrete RV 
Continuous RV 
 Let X and Y be two RVs. The CONDITIONAL DISTRIBUTION of Y given X=x is,
X and Y are two RVs. The CONDITIONAL DISTRIBUTION of X given Y=y is,
Q3.8: For questions Q3.6, calculate the marginal distribution, g(x) and h(y) of the RVs, X, and
Y, and f(1|1).
For questions Q3.7, calculate the marginal distribution, g(x) and h(y) of the RVs, X, and Y and
f(0.5|0.5) (Hint: h(0.5)= 
).
Q3.9: Two refills of an HP color printer are randomly selected from a box, which contains 3
blue refills, 2 red refills, and 3 green refills. Let X and Y be the number of blue and green refills,
which are selected, respectively, and at random.

Stat & Prob. for Engineers_ENSIT2025/2026 Ali TRABELSI, Dr., Eng., 5
Q1. Find the J.P.F, f(x, y).
Q2. Prob. {(X, Y) ϵ A, x+y ≤ 1}.
Q3. Find f(x|1) for x=0,1 and 2.
Q3.10: The J.P.D for two R.Vs X and Y is given by
Q1. Find the marginal density functions g(x) and h(y). Find f(y|x)
Q2. Find f(Y>½|X=¼).
Hint:  
E. STATISTICAL INDEPENDENCE
Given X, Y two RVs having f(x, y), g(x), and h(y) as joint prob. dist., marginal dist. in X and
marginal dist. in Y, resp.; X and Y are STATISTICALLY INDEPENDENT if and only if,
Q3.11: Are the RVs X and Y of Q3.6 and Q3.7 statistically independent?
Hint:  
F. GENERALIZATION
 Let f(x1, x2, .., xn) be the J.P.F of the RVs, X1, X2, .., Xn
Marginal distribution of X1
Joint marginal distribution of X1 and X2