This review covers all lessons in this topic: discrete random variables and probability functions, mean/variance/standard deviation, Bernoulli distributions, and binomial distributions. Questions are mixed in difficulty.
Review Questions
A discrete random variable X has the following probability distribution:
x
1
2
3
4
P(X=x)
0.1
0.3
0.4
0.2
Verify this is a valid probability distribution and find P(X ≥ 3).
A bag contains 3 red and 2 blue balls. One ball is drawn. Let X = 1 if red, X = 0 if blue. Write the probability distribution of X and state what type of random variable this is.
Find the value of k that makes the following a valid probability distribution, where P(X=x) = kx for x = 1, 2, 3, 4, 5.
For a Bernoulli random variable with p = 0.35, find E(X), Var(X) and SD(X).
A discrete random variable X has E(X) = 4 and E(X²) = 20. Find Var(X) and SD(X).
Let X ~ Bin(8, 0.3). Find: (a) P(X = 2) (b) P(X ≤ 1) (c) E(X) and SD(X).
A fair coin is tossed 6 times. Let X = number of heads. Find P(X = 4) and P(at least 4 heads).
A random variable X has the probability distribution:
x
0
1
2
3
P(X=x)
0.2
0.4
0.3
0.1
Calculate E(X) and Var(X).
In a multiple-choice test, each question has 4 options and one correct answer. If a student randomly guesses on 10 questions, find the probability they get exactly 3 correct. What is the expected number of correct answers?
A factory produces items of which 5% are defective. A quality inspector samples 20 items. Let X = number of defective items.
(a) State the distribution of X and its parameters.
(b) Find the probability that at most 1 item is defective.
(c) Find E(X) and interpret this in context.
The probability that a basketball player scores on any given free throw is 0.7. She takes 5 free throws. Find the probability she scores on more than 3 throws.
Let X ~ Bin(n, p) with E(X) = 6 and Var(X) = 4.2. Find n and p.
A Bernoulli trial succeeds with probability p. If the mean of the distribution is 0.4, find:
(a) The value of p
(b) P(X = 1)
(c) The variance
A spinner has outcomes 1, 2, 3, 4, 5 each equally likely. X is the outcome of one spin. Calculate E(X), E(X²), Var(X) and SD(X).
A school estimates that 60% of students bring a lunch from home. A class of 12 students is selected. Let X = number of students who bring lunch.
(a) Find P(X = 7).
(b) Find P(5 ≤ X ≤ 8).
(c) Find the most likely number of students who bring lunch (the mode).