Stats Chapter 5 - Continuous Random Variables

What is a continuous random variable?

A variable whose values are measured rather than counted.

What is a discrete random variable?

A variable whose values are counted.

What does probability represent on a continuous density graph?

Area under the curve.

What does the total area under a probability density function equal?

1.

What does P(X = c) equal for a continuous random variable?

0.

Why is P(X = c) = 0 for a continuous variable?

A single point has zero width and therefore zero area.

For a continuous variable, P(X < c) equals what?

P(X ≤ c).

For a continuous variable, P(X > c) equals what?

P(X ≥ c).

What is the probability outside a distribution's possible range?

0.

What is the probability over the entire possible range?

1.

What does the CDF represent?

The probability/area to the left of x.

What does P(X > x) equal using the CDF?

1 − [P(X ≤ x)]

What does P(a < X < b) equal using CDF values?

P(X < b) − P(X < a)

How do you recognize a uniform distribution from its graph?

It has a rectangular, constant-height shape.

How do you recognize an exponential distribution from its graph?

It is a decreasing curve starting at x = 0.

How do you recognize a normal distribution from its graph?

It is a symmetric bell-shaped curve.

What does equally likely mean for a uniform distribution?

Every value in the interval is equally likely.

What is the notation for a uniform distribution?

X ~ U(a,b).

In X ~ U(a,b), what is a?

The lower/minimum value.

In X ~ U(a,b), what is b?

The upper/maximum value.

Why is the uniform PDF height 1 ÷ (b − a)?

Rectangle area must equal 1.

How do you find uniform probability from a graph?

Calculate the shaded area: base × height.

In the percentile formula, what does p represent?

The desired percentile written as a decimal.

How do you solve a uniform percentile problem?

Set the probability equal to p and solve for k.

What is the mean of X ~ U(0,12)?

6.

What is the PDF height of X ~ U(0,12)?

1/12.

What does the range/support of a random variable mean?

The values the variable can possibly take.

What happens if a probability interval is completely outside the range?

The probability is 0.

How do you graph P(X > 9) for X ~ U(0,12)?

Shade the region from 9 to 12.

What does conditional probability mean?

The probability of an event given another condition is known.

What changes when calculating conditional probability?

The given condition becomes the new sample space.

For X ~ U(a,b), what is P(X > c | X > d) when c > d?

( b − c )/( b − d ).

What is the main idea behind a uniform conditional probability?

Restrict the range first, then calculate probability within it.

What is the exponential distribution used to model?

Time until an event occurs.

What type of variable is exponential?

Continuous.

What does Poisson measure?

The number of events in a time period.

What does exponential measure?

The time between events.

What is the easiest way to remember Poisson vs exponential?

Poisson = how many; exponential = how long.

What does λ represent in an exponential distribution?

The rate at which events occur.

What is the relationship between λ and μ?

λ = 1/μ and μ = 1/λ.

What is the exponential mean?

μ = 1/λ.

What is the exponential standard deviation?

σ = 1/λ = μ.

What formula should you use for "less than" in an exponential problem?

P(X < x) = 1 − [e^(−λx)]

What formula should you use for "more than" in an exponential problem?

P(X > x) = e^(−λx).

What formula should you use for "between" in an exponential problem?

[e^(−λa)] − [e^(−λb)]

What exponential wording usually means P(X < x)?

• Within

• before

• less than

• at most

• no more than

What exponential wording usually means P(X > x)?

• More than

• longer than

• after

• least

If the average waiting time is 4 minutes, what is μ?

4 minutes.

If μ = 4 minutes, what is λ?

0.25 per minute.

If λ = 0.5 per minute, what is μ?

2 minutes.

If 30 customers arrive per hour, what is λ?

30 per hour.

If 30 customers arrive per hour, what is the average time between arrivals?

2 minutes.

Why must λ and x use matching units?

The exponent λx must be unitless.

If λ is per hour, what unit must x use?

Hours.

What should you do if λ is per minute but x is given in seconds?

Convert x to minutes first.

What is the shape of an exponential distribution?

Decreasing and right-skewed.

Where is an exponential PDF highest?

At x = 0.

What is f(0) for an exponential distribution?

λ.

What does a right-skewed exponential distribution imply?

Many smaller values and fewer larger values.

What is the memoryless property?

Past waiting time does not affect future waiting time.

How do you solve P(X > 10 | X > 5) for an exponential?

Use P(X > 5).

What does the memoryless property mean in words?

The future depends only on future waiting time, not past waiting.

How does the exponential mean compare with its median?

Mean > median.

Why is the exponential mean greater than its median?

The distribution is right-skewed.

How do you find an exponential percentile?

Set 1 − e^(−λx) equal to the desired probability and solve for x.

How do you find an exponential median?

Set P(X < x) = 0.50 and solve for x.

What does μ represent in a Poisson distribution?

The average number of events.

What does μ represent in an exponential distribution?

The average waiting time.

What is the relationship between Poisson and exponential distributions?

Poisson counts events; exponential measures time between events.