Lecture #5: Discrete Probability Distributions

Motivating Example

Example: I offer a lottery game for $5 a play. I spin a spinner with 1-10 on it, evenly spread out. If it lands on 1-9, I keep your money. If it lands on 10, you win $100. Let \(X\) be the payout you receive from this game.

Bernoulli Distribution

These kinds of random variables can be described using a Bernoulli distribution.

PMF of the Bernoulli Distribution

The probability mass function of the Bernoulli distribution with success probability \(p\) is: \[ p_X(x) = P(X = x) = \begin{cases} p & \text{for } x = 1 \\ 1 - p & \text{for } x = 0 \end{cases} \]

Expected Value and Variance of the Bernoulli Distribution

The expected value of a \(\text{Bernoulli}(p)\) distribution is: \[ E(X) = p. \]

Example: Bernoulli Distribution

Example: You flip a biased coin, landing on heads 65% of the time. If the coin lands on heads, you make $5, and if it lands on tails, you get nothing. Let \(X\) be the payout of the game.

1. What is the distribution of \(X\)?
2. What is the expected payout?
3. What is the variance and standard deviation of the payouts?

The Binomial Distribution

Suppose instead of flipping a single coin, we flip it 10 times, and record the number of heads, and call that variable \(X\). How is \(X\) distributed?

The Binomial Distribution

The binomial distribution is defined as the sum of \(n\) independent and identically distributed Bernoulli random variables with success probability \(p\).

PMF of the Binomial Distribution

The probability mass function of a binomially distributed random variable \(X\) is: \[ p_X(x) = P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}, \] for \(x \in \{ 0, 1, 2, \dots, n \}\).

Cumulative Distribution Function

There are times when the probability mass function isn’t the most useful way to frame a problem.

Binomial Distribution Example

Example:

The probability that a patient recovers from a stomach disease is \(.8\). Suppose \(20\) people are known to have contracted this disease. What is the probability that:

1. exactly 14 recover?
2. at least 10 recover?
3. at least 14 but not more than 18 recover?
4. at most 16 recover?

Expected Value and Variance of the Binomial Distribution

The expected value of a \(\text{Binomial}(n, p)\) distribution is: \[ E(X) = np. \]

Binomial Distribution Example

Example:

An oil exploration firm is formed with enough capital to finance ten explorations. The probability of a particular exploration being successful is \(.1\). Assume the explorations are independent. Find the mean and variance of the number of successful explorations.

The Geometric Distribution

Let’s consider a new game: We are going to flip the coin until we get a tails. We will pay out $1 for each coin flip it takes until we get a tails.

• If the first coin flip is a tails (\(T\)), you get $1.
• If the first coin flip is a heads and the second coin flip is a tails (\(HT\)), you get $2.
• If the first two coin flips are heads and the third is a tails (\(HHT\)), you get $3.
• If the first three coin flips are heads and the fourth is a tails (\(HHHT\)), you get $4.
• And so on and so on…

The Geometric Distribution

Let \(X\) be the payout of this game.

• What is the sample space for this game?
• What might the probability mass function look like?

PMF of the Geometric Distribution

The probability mass function of a random variable \(X\) with a geometric distribution is: \[ p_X(x) = P(X = x) = (1-p)^{x-1} p, \]

for \(x \in \{ 1, 2, 3, \dots \}\).

Geometric Distribution Example

Example:

Consider the game from earlier. Suppose the coin lands on tails \(70\%\) of the time. What is the probability that:

1. the game ends on the third coin flip?
2. the game lasts \(2\) or fewer rounds?
3. the game lasts more than two rounds?
4. the game lasts more than four rounds, given that it has lasted more than two rounds?

Expected Value and Variance of the Geometric Distribution

The expected value of a \(\text{Geometric}(p)\) distribution is: \[ E(X) = \frac{1}{p}. \]

Geometric Distribution Example

Example:

Of a population of consumers, \(60\%\) are reputed to prefer a particular brand, \(A\), of toothpaste. If a group of randomly selected consumers is interviewed, what is the mean and standard deviation of the number of consumers that must be interviewed until someone who doesn’t prefer brand \(A\) is found?

The Negative Binomial Distribution

We can further generalize the idea behind the geometric distribution. What if instead of waiting until the first success, we wait until the second success, or the third success, or the \(r^{th}\) success?

• What is the sample space for this distribution?
• What might the probability mass function look like?

PMF of the Negative Binomial Distribution

The probability mass function of a random variable \(X\) with a negative binomial distribution is: \[ p_X(x) = P(X = x) = \binom{x - 1}{r - 1} (1-p)^{x-r} p^r, \]

for \(x \in \{ r, r+1, r+2, \dots \}\).

Expected Value and Variance of the Negative Binomial Distribution

The expected value of a \(\text{NB}(r, p)\) distribution is: \[ E(X) = \frac{r}{p}. \]

Negative Binomial Example

A geological study indicates that an exploratory oil well should strike oil with probability \(.2\).

1. What is the probability that the third strike comes on the seventh well drilled?
2. What assumptions did you make to obtain the answer to part (1)?
3. Find the mean and variance of the number of wells that must be drilled if the company wants to set up three producing wells.

The Hypergeometric Distribution

Consider an urn with 8 red balls and 7 black balls in it. If I draw 5 balls from the urn, without replacement, what is the probability that the majority of the balls we pulled out are black?

Let \(Y\) be the number of black balls drawn from the urn in the five balls drawn.

• What is the sample space for this experiment?
• What might the probability mass function look like?

The Hypergeometric Distribution

PMF of the Hypergeometric Distribution

The probability mass function of a random variable \(Y\) with a hypergeometric distribution is: \[ p_Y(y) = P(Y = y) = \frac{\binom{r}{y} \binom{N-r}{n-y}}{\binom{N}{n}}, \]

where \(y\) is an integer \(\{0,\, 1,\, 2,\, \dots,\, n\}\), subject to the restrictions \(y \le r\) and \(n − y \le N − r\).

Expected Value and Variance of the Hypergeometric Distribution

The expected value of a \(\text{Hypergeometric}(N, r, n)\) distribution is: \[ E(Y) = \frac{nr}{N}. \]

Hypergeometric Distribution Example

Example:

An urn contains ten marbles, of which five are green, two are blue, and three are red. Three marbles are to be drawn from the urn, one at a time without replacement.

1. What is the probability that all three marbles drawn will be green?
2. What is the probability that at most one marble drawn will be green?
3. What is the mean of the number of green marbles drawn?
4. What is the variance of the number of green marbles drawn?

The Poisson Distribution

The manager at McDonald’s called me to ask for some help modelling his sales. In particular, he wants to predict the number of customers that will come through the drive-thru in an hour, so he can properly staff the window.

Let \(X\) be the number of customers that come through the drive-thru in an hour.

• What is the sample space for this experiment?
• What might the probability mass function look like?

The Poisson Distribution

The Poisson distribution expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

PMF of the Poisson Distribution

The probability mass function of a random variable \(X\) with a Poisson distribution is: \[ p_X(x) = P(X = x) = \frac{\lambda^x \, e^{-\lambda}}{x!}, \]

where \(x\) is a non-negative integer and \(\lambda > 0\).

Expected Value and Variance of the Poisson Distribution

The expected value of a \(\text{Poisson}(\lambda)\) distribution is: \[ E(X) = \lambda. \]

Poisson Distribution Example

Example:

A certain type of tree has seedlings randomly dispersed in a large area, with the mean density of seedlings being approximately five per square yard.

1. What is the probability that a single one square yard region will not have any seedlings?
2. If a forester randomly locates ten 1-square-yard sampling regions in the area, find the probability that none of the regions will contain seedlings.