Question

. Suppose X ∼ Unif(−1, 3). Find the probabilities of the following events, both by hand calculation and with R’s punif function. (a) (X ≤ 2) (b) (X ≥ 1) (c) (−0.5 < X < 1.5) (d) (X = 0)

Answer 1

Answer:

a) $P(X \leq 2) = \frac{2+1}{3+1}= 0.75$

b) $P(X \geq 1) = 1- P(X$

c) $P(-0.5$

d) $f(x) = \frac{1}{3+1}= 0.25$

Step-by-step explanation:

Let X the random variable of interest and we know that the distribution is given by:

$X \sim Unif (a= -1, b=3)$

And for this problem we can use the cumulative distribution function in order to solve the items:

$F(x) =\frac{x-a}{b-a}, a\leq x \leq b$

Part a

We want to find this probability:

$P(X \leq 2) = \frac{2+1}{3+1}= 0.75$

Part b

$P(X \geq 1) = 1- P(X$

Part c

$P(-0.5$

And we can calculate the probability with this difference:

$P(-0.5$

Part d

Since we have a continuous distribution the the probability for an unique value would be:

$f(x) = \frac{1}{3+1}= 0.25$