Question

Suppose the cumulative distribution function of the random variable X is F(x) = 0 when x<-2 , F(x) = .25x + .5 when -2 <= x < 2 and F(x) = 1 when 2<=x (<= means greater than or equal). Determine the following a. P(X<1.8) b. P(X>-1.5) c. P(X<-2) d. P(-1

Answer 1

To calculate probabilities using the CDF of a variable X, plug the value into the CDF's equation. For the interval -2 <= x < 2, we use F(x) = 0.25x + 0.5. For values outside this range, we use the given values in the CDF definition.

Step-by-step explanation:

The cumulative distribution function (CDF) of a random variable X provides the probability that X will take on a value less than or equal to x. Here's how we can use it to solve the problems indicated:

1. To find P(X<1.8), use the function for the CDF in the interval -2 ≤ x < 2: F(x) = 0.25x + 0.5. Plugging in 1.8 gives F(1.8) = 0.25(1.8) + 0.5 = 0.95.
2. The probability that X > -1.5 can be found as follows: P(X > -1.5) = 1 − P(X ≤ -1.5) = 1 − [0.25(-1.5) + 0.5] = 1 − 0.125 = 0.875.
3. P(X < -2) is provided directly by the CDF: Since X < -2 is outside the defined range for the variable part of the CDF, P(X < -2) = 0.
4. To calculate P(-1.5 < X < 1.5), use the CDF to get P(X ≤ 1.5) − P(X ≤ -1.5): [0.25(1.5) + 0.5] − [0.25(-1.5) + 0.5] = 0.875 − 0.375 = 0.5.