For each probability and percentile problem, draw the picture.
Let X ~ Exp(0.2).
Part (g)
Sketch a new graph, shade the area corresponding to the 40th percentile, and find the value. (Round your answer to two decimal places.)
The exponential distribution is a continuous probability distribution with a graph that starts at the origin and decays exponentially to the right. Since the rate parameter is 0.2, the curve is steeper compared to the exponential distribution with a smaller rate parameter.
To find the 40th percentile, we need to find the value of x for which 40% of the distribution lies to the left of x.
To sketch the graph, we start at the origin and draw a curve that decays exponentially to the right. We then shade the area under the curve to the left of the 40th percentile.
Let's say the x-axis represents the possible values of X, and the y-axis represents the probability density function (PDF) of X. The shaded area represents the cumulative distribution function (CDF), which gives the probability that X is less than or equal to a specific value.
The 40th percentile represents the value of x for which the cumulative probability is 0.4.
To find this value, we can use the exponential distribution's cumulative distribution function (CDF) formula:
CDF(x) = 1 - e^(-λx)
where λ = 0.2 (rate parameter) and CDF(x) is the cumulative probability of X being less than or equal to x.
Now we can solve for x:
0.4 = 1 - e^(-0.2x)
Rearranging the equation:
e^(-0.2x) = 1 - 0.4
e^(-0.2x) = 0.6
Taking the natural logarithm (ln) of both sides:
-0.2x = ln(0.6)
Dividing both sides by -0.2:
x ≈ -ln(0.6)/0.2 ≈ 3.48 (rounded to two decimal places)
So the 40th percentile of the exponential distribution with a rate parameter of 0.2 is approximately 3.48.
The final sketch would show the exponential curve, shading the area to the left of x = 3.48, indicating the 40th percentile.