2.1.4 Mixed Distributions Flashcards
(16 cards)
What is a mixed distribution?
A distribution made from both discrete and continuous components (or multiple of each type).
What are the three types of mixed distributions?
(1) Multiple discrete, (2) multiple continuous, or (3) a mix of discrete and continuous distributions.
Why can’t the PDF or PMF alone describe a mixed distribution?
Because the probability function includes both types — it’s not fully discrete or fully continuous.
What happens when you integrate the continuous part of a mixed distribution and it doesn’t sum to 1?
It means the distribution is incomplete — the rest of the probability is in discrete components.
In a mixed distribution, why might it be easier to handle the distribution ‘as a whole’?
Because the components aren’t cleanly separated; treating them together avoids unnecessary decomposition.
What is one way to visually identify a mixed distribution using a CDF?
Look for jumps (discontinuities) in the CDF — each jump indicates a discrete probability.
If the CDF of a distribution jumps at x = 2 by 0.5, what does that mean?
P(X = 2) = 0.5 — there is a discrete component at x = 2.
How can you get the PDF portion of a mixed distribution from the CDF?
By taking the derivative of the continuous portions of the CDF.
What does it mean if the total area under the PDF of a distribution is less than 1?
The missing probability is likely in discrete values — it’s a mixed distribution.
What’s a quick check to confirm if a distribution is mixed using the CDF?
Check if the CDF has discontinuous jumps and a continuous slope — that’s mixed.
When is it hard to tell a distribution is mixed?
When you’re given only a CDF or survival function — you must check endpoints for jumps.
What does a jump in a CDF graph indicate?
A discrete probability at that point.
What must the total probability (discrete + continuous) equal for a valid probability function?
1 — always.
How can you express a mixed distribution’s probability function?
Combine the discrete probabilities and the continuous PDF into one piecewise expression.
Why does a piecewise-defined function often indicate a mixed distribution?
Because its definition can change between discrete outcomes and continuous ranges.
What is the key skill when working with mixed distributions in problems?
Being able to break it down case by case or handle it intuitively as one whole object.
