chapter5 Flashcards
What is a random variable?
A random variable is a variable that takes numerical values based on the outcomes of a random experiment.
What are the two types of random variables?
Discrete Random Variable: Takes a countable number of values (e.g., number of steps taken).
Continuous Random Variable: Takes any value within a range (e.g., time spent).
What is the probability distribution of a discrete random variable?
A table or graph showing the probabilities of all possible outcomes.
Rule 1: P(x)≥0
Rule 2: ΣP(x)=1
How is the expected value (mean) of a discrete random variable calculated?
The expected value is the weighted average of all possible values:
μ=E(x)=ΣxP(x)
What is a binomial experiment?
A binomial experiment has:
Fixed number of trials (n)
Two outcomes (success/failure)
Constant probability of success (p)
Independent trials.
What is the formula for a binomial probability?
P(x)=(n choose x)p^x(1−p)^(n−x)
Where:
n: Number of trials
x: Number of successes
p: Probability of success
What is a Poisson distribution?
A distribution for the number of events occurring in a fixed interval of time or space when:
Events occur independently.
The average rate (λ) is constant.
What is the formula for a Poisson probability?
P(x)=e^−λ(λ^x)/x!
Where:
λ: Mean number of events
x: Number of events
e: Base of natural logarithms (≈2.718)
How are binomial and Poisson distributions different?
Binomial: Fixed number of trials, two outcomes, probability remains constant.
Poisson: Models rare events over continuous time/space, no fixed trials.
How do you calculate variance and standard deviation for a discrete random variable?
Variance: σ²=Σ(x−μ)²P(x)
Standard Deviation: σ=√σ²
What are some examples of discrete random variables?
Number of patients arriving at a clinic.
Number of tails in a coin toss.
Number of defective items in a batch.
What keywords in questions help identify a binomial distribution?
“Fixed number of trials”
“Two outcomes: success or failure”
“Probability remains constant”
“Independent trials”
What keywords in questions help identify a Poisson distribution?
“Events per unit of time/space”
“Rare events”
“Independent occurrences”
“Rate of occurrence is constant”
What is the Empirical Rule for probability distributions?
For a mean (μ) and standard deviation (σ):
P(μ−σ≤x≤μ+σ)≈68%
P(μ−2σ≤x≤μ+2σ)≈95%
P(μ−3σ≤x≤μ+3σ)≈99.7%
What is the cumulative distribution function (CDF) for a discrete random variable?
The CDF gives the probability that the random variable X is less than or equal to a specific value x:
F(x)=P(X≤x)=ΣP(X=k) for k≤x
What are some real-life applications of the binomial distribution?
Flipping a coin a fixed number of times.
Testing a batch of products for defects.
Conducting surveys with yes/no responses.
What are some real-life applications of the Poisson distribution?
Number of calls received by a call center per hour.
Number of accidents at a traffic intersection in a day.
Number of printing errors in a book.
What is the variance of a binomial distribution?
The variance of a binomial distribution is:
σ²=n⋅p⋅(1−p)
What is the variance of a Poisson distribution?
The variance of a Poisson distribution equals its mean:
σ²=λ
How do you recognize if a problem involves a discrete random variable?
The variable involves countable outcomes (e.g., 0, 1, 2, …).
The probability of each outcome can be listed.
Examples: Number of students in a class, dice rolls.
What is the shape of a binomial probability distribution?
Symmetric: When p=0.5 and n is large.
Skewed: When p is closer to 0 or 1.
How is the Poisson distribution derived from the binomial distribution?
The Poisson distribution is a limiting case of the binomial distribution when:
n→∞ (large number of trials).
p→0 (small probability of success).
λ=n⋅p (mean remains constant).
What is the relationship between mean and variance in the Poisson distribution?
In a Poisson distribution, the mean and variance are equal:
μ=σ²=λ
What are the key characteristics of a probability mass function (PMF)?
Assigns probabilities to each value of a discrete random variable.
The sum of all probabilities equals 1.
Probabilities are non-negative.
What is the difference between PMF and PDF?
PMF (Probability Mass Function): Used for discrete random variables, assigns probability to exact values.
PDF (Probability Density Function): Used for continuous random variables, represents probabilities over intervals.
How does the law of large numbers apply to random variables?
As the number of trials increases, the sample mean of a random variable approaches its expected value.
What is the central limit theorem?
For large sample sizes, the sampling distribution of the sample mean approaches a normal distribution, regardless of the population’s original distribution.
How do you find the mode of a discrete random variable?
The mode is the value of X that has the highest probability P(X=x).
What is the standard deviation of a binomial distribution?
The standard deviation of a binomial distribution is:
σ=√(n⋅p⋅(1−p))
How can you identify if a problem involves Poisson distribution?
Look for:
Number of events in a fixed interval.
Rare or infrequent events.
No upper limit on the number of occurrences.
