**Probability mass functions (p.m.f. s)**

To calculate

*P*(

*X*=

*x*) for a discrete distribution, you can use the probability mass function (p.m.f.).

In R, these usually have function names beginning with 'd'.

For example, to calculate the probability of obtaining 7 'Yes' responses and 3 'No' responses from a sample of 10 people questioned, if the probability of a 'Yes' is 1/3, we use the p.m.f. for a Binomial distribution:

> dbinom(7, size=10, prob=1/3)

[1] 0.01625768

**Probability density functions (p.d.f. s)**

To calculate P(

*X*<=

*x*) for a continuous distribution, you can use the probability density function (p.d.f.). As for p.m.f.s, in R these usually have function names beginning with 'd'.

For example, for a Normal distribution with mean 100 and standard deviation 15 (variance 225), to calculate the probability of observing a score of 110 or higher, we type:

> 1 - pnorm(110, mean=100, sd=15)

0.2524925

We get the same answer by typing:

> pnorm(110, mean=100, sd=15, lower.tail=FALSE)

0.2524925

**Cumulative distribution functions (c.d.f. s)**

To calculate

*P*(

*X*<=

*x*), we can use cumulative distribution functions.

In R, these usually have function names beginning with 'p'.

For example, for an exponential distribution with rate parameter 0.25, to calculate

*P*(

*X*<= 2), we type:

> pexp(2, rate = 0.25)

[1] 0.3934693

We can use the cumulative distribution function to find the probability that an interval will lie in a given range, for example, to calculate

*P*(5 <=

*X*<= 10), we type (again using an exponential distribution):

> pexp(10, rate = 0.25) - pexp(5, rate = 0.25)

[1] 0.2044198

Similarly, if the probability of getting one answer right by chance in a multiple choice exam is 1/5, the probability of getting ten or more answers right (out of twenty questions) by chance is (using a Binomial distribution):

> 1 - pbinom(9, size=20, prob=1/5)

0.002594827

We also get the same answer if we type:

> pbinom(9, size=20, prob=1/5, lower.tail=FALSE)

0.002594827

Another example is using a Geometric distribution to find the probability that you need to roll a die at at most 3 times to obtain a six:

> pgeom(2, prob=1/6)

0.4212963

[Note: the pgeom() function in R takes as its argument the number of failures before the first success.]

## No comments:

Post a Comment