The notation X ~ U(a, b) names the uniform distribution, whose density is flat at height 1/(b - a).
Missing Word
uniform
The uniform distribution gives equal density to every value between a and b. Its graph is a rectangle of height 1/(b - a), so equal-length intervals have equal probability.
Category: Uniform Distribution
Question 2
The time until the next phone call, with density f(x) = m e^(-mx), follows the exponential distribution.
Missing Word
exponential
The exponential distribution models waiting times and lifetimes. Its density starts at height m at x = 0 and decreases, and its mean is 1/m.
Category: Exponential Distribution
Question 3
A waiting-time distribution is called memoryless when the chance of waiting t more minutes is the same no matter how long one has already waited.
Missing Word
memoryless
Memoryless means P(X > r + t | X > r) = P(X > t). The exponential distribution has this property, so time already spent waiting does not change the chance of the remaining wait.
Category: Exponential Distribution
Question 4
If times between independent events have density m e^(-mx), the count of events per unit of time follows a Poisson distribution with an average of m.
Missing Word
Poisson
The Poisson distribution counts events in a fixed interval. When the gaps between independent events are exponential with mean 1/m, the number of events per unit time is Poisson with mean m.
Category: Poisson Connection
Question 5
When every value from a to b is equally likely, the region under the density forms a rectangle of height 1/(b - a).
Missing Word
rectangle
A uniform density is a constant, so the region beneath it is a rectangle with base b - a and height 1/(b - a). Its area, base times height, is exactly 1.
Category: Uniform Distribution
Question 6
The median of X is the value c with P(X < c) = 0.5.
Missing Word
median
The median is the 50th percentile, the point that splits the probability in half. For a skewed distribution such as the exponential, it differs from the mean.
Category: Percentiles
Question 7
For X ~ U(0, 20), the value 18 is the 90th percentile because P(X < 18) = 0.90.
Missing Word
percentile
A percentile is the value with a given proportion of probability at or below it. For a uniform variable, the kth percentile is a + p(b - a), here 0 + 0.90(20) = 18.
Category: Percentiles
Question 8
For X ~ U(a, b), the variance is (b - a)^2/12.
Missing Word
variance
The variance measures spread in squared units. For a uniform distribution it equals (b - a)^2/12, and taking its square root gives the standard deviation.
Category: Uniform Distribution
Question 9
The identity P(X > x) = 1 - P(X < x) follows from the complement rule of probability.
Missing Word
complement
The complement of an event is everything outside it, and the two probabilities add to 1. For a continuous variable, the area to the right of x is 1 minus the area to the left.
Category: Probability as Area
Question 10
A density that peaks at x = 0 and trails off in a long tail toward large values is skewed to the right.
Missing Word
skewed
A skewed distribution has one tail longer than the other. The exponential density is the classic right-skewed shape: it is highest at 0 with a long tail to the right.
Category: Exponential Distribution
Question 11
A random variable that takes only separate, countable values, such as the number of calls in an hour, is discrete.
Missing Word
discrete
A discrete random variable has countable values, and its probabilities can be listed one value at a time. Measured quantities like time are not discrete.
Category: Continuous Variables
Question 12
A random variable measured on a scale, such as a weight that can be any value from 50 to 80 kilograms, is continuous.
Missing Word
continuous
A continuous random variable can take any value in an interval. Because there are infinitely many possible values, single values have probability 0 and probabilities come from areas.
Category: Continuous Variables
Question 13
Finding P(X > 12 | X > 8) means computing a conditional probability.
Missing Word
conditional
A conditional probability is the chance of one event given that another has occurred: the probability of both events divided by the probability of the given event. For a uniform variable it can be found by shrinking the range to the given condition.
Category: Probability as Area
Question 14
For a random variable measured on a scale, each probability is found as an area under the graph of its density f(x).
Missing Word
area
For continuous variables, probability equals the area under the density curve over an interval. The total area under the curve is 1.
Category: Probability as Area
Question 15
A density that is flat from a to b has its axis of symmetry at the midpoint (a + b)/2.
Missing Word
symmetry
Symmetry means the graph is a mirror image on either side of a center line. A flat rectangle from a to b mirrors itself about (a + b)/2, which is why the mean of a uniform distribution sits exactly at that midpoint.
Category: Uniform Distribution