A random variable is a rule that assigns a number to the outcome of a random experiment.
In simpler terms:
- You perform a random experiment (like flipping a coin or rolling a die).
- Each possible outcome is assigned a numerical value.
- That numerical value is the random variable.
Problem 1: Let X be a random variable with the following probability mass function (pmf): P(X = 1) = 0.2, P(X = 2) = 0.5, and P(X = 3) = 0.3. Find the expected value E(X).
Solution:
E(X) = 1×0.2 + 2×0.5 + 3×0.3
= 0.2 + 1 + 0.9 = 2.1
Problem 2: A fair six-sided die is rolled. Let X be the outcome of the roll. Find the variance Var(X).
Solution:
E(X) = 1 + 2 + 3 + 4 + 5 + 6/6
= 3.5
E(X2) = 12 + 22 + 32 + 42 + 52 + 62
= 15.667
Var(X) = E(X2) - (E(X))2
= 15.667 -3.52
= 2.9167
Problem 3: Let Z be a random variable with the following probability mass function (pmf): P(Z = -1) = 0.3, P(Z = 0) = 0.4, and P(Z = 1) = 0.3. Find the expected value E(Z)and the variance Var(Z).
Solution:
E(Z) = (-1)×0.3 + 0×0.4 + 1×0.3
= 0
E(Z2) = (-1)2×0.3 + 02×0.4 + 12×0.3
= 0.6
Var(Z) = E(Z2)-(E(Z))2 = 0.6-0
= 0.6
Problem 4: A random variable W follows a uniform distribution on the interval [2, 4]. Find the expected value E(W) and the variance Var(W).
Solution:
E(W) = 2 + 4/2
= 3
Var(W) = (4-2)2 /12 = 4/12
= 1/3
Problem 5: Let X be a Bernoulli random variable with parameter p = 0.4. Find the expected value E(X) and the variance Var(X).
Solution:
E(X) = p = 0.4
Var(X) = p(1-p) = 0.4⋅(0.6)
= 0.24
Problem 6: Suppose X and Y are independent random variables with X following a normal distribution with mean 2 and variance 1, and Y following a normal distribution with mean 3 and variance 4. Find the distribution of Z = X + Y.
Solution:
Z∼N(μX + μY,σX2 + σY2)
= N(2 + 3,1 + 4)
= N(5,5)
Problem 7: A random variable X follows a geometric distribution with parameter p = 0.3. Find the expected value E(X).
Solution:
E(X) = 1/p
= 1/0.3
= 10/3≈3.33
Problem 8: If Z is a standard normal random variable, find P(Z > 1.96).
Solution:
Using standard normal distribution table:
P(Z > 1.96)
= 1 - P(Z ≤ 1.96)
= 1-0.975 = 0.025
Problem 9: Let X be a random variable with the following probability mass function (pmf): P(X = 0) = 0.4, P(X = 1) = 0.3, and P(X = 2) = 0.3. Find the expected value E(X) and the variance Var(X).
Solution:
E(X) = 0⋅0.4 + 1⋅0.3 + 2⋅0.3
= 0.9
E(X2) = 02×0.4 + 12×0.3 + 22×0.3 = 0 + 0.3 + 1.2
= 1.5
Var(X) = E(X2)-(E(X))2 = 1.5-0.92 = 1.5-0.81
= 0.69
Problem 10: If Z is a standard normal random variable, find P(-2 ≤ Z ≤ 2).
Solution:
Using standard normal distribution table
P(-2 ≤ Z ≤ 2 )
= P(Z ≤ 2) - P(Z ≤ -2)
= 0.9772 - 0.0228
= 0.9544
Practice Problems
Q1. Let X be a discrete random variable with the following pmf: P(X = 1) = 0.1, P(X = 2) = 0.3, P(X = 3) = 0.4, and P(X = 4) = 0.2. Find the expected value E(X).
Q2. A random variable Y follows a uniform distribution on the interval [1, 5]. Find the probability that Y is greater than 3.
Q3. If Z is an exponential random variable with rate parameter λ = 2, find the probability that Z is less than 1.
Q4. Let X be a normal random variable with mean μ = 4 and variance σ2 = 9. Find P(1 ≤ X ≤ 7).
Q5. A fair coin is flipped 10 times. Let X be the number of heads obtained. Find the probability that X equals 5.
Q6. Let Y be a continuous random variable with the pdf fY(y) = 1/2sin(y) for 0 ≤ y ≤ π. Find the expected value E(Y).
Q7. A random variable Z follows a Poisson distribution with parameter λ = 6. Find the probability that Z equals 4.
Q8. Let X be a Bernoulli random variable with parameter p = 0.7. Find the expected value E(X) and the variance Var(X).
Q9. A random variable X follows a binomial distribution with parameters n = 8 and p = 0.4. Find the probability that X is at least 6.
Q10. If Z is a standard normal random variable, find P(Z ≤ -1.28).
