Question

Let X be a normal random variable with ?=−10 and ?=2. Let Z be a standard normal random variable. Draw density plots for both random variables on the same graph. You will want an x-axis that goes from around -20 to around 5. Your y-axis will start at zero and will need go high enough to cover the highest density. Recall that the density of a normal random variable at the point ? with mean ? and standard deviation ? is 1?2?√?−12(?−??)2. Your plots do not need to be perfect but try to make them reasonably accurate. It helps to compute the density at several different values for each random variable (the mean should definitely be one of these points). Include axis tick marks for both axes and clearly label the curves, indicating which curve belongs to the density function of which random variable (X or Z). Also, on the x-axis, clearly label the centers of symmetry. On the y-axis, clearly label the peaks of the curves. Shade the region -12 < X < -6 under the density curve for X. Also shade the region -1 < Z < 2. If you have drawn the graph correctly, these two regions should be about the same area on your page. Why is this? Explain, using the concept of probability and what you know about normal random variables (1-2 sentences). Give a rough estimate of the area of this region (you don't need to use math). What is P(-12 < X < -6)? What is P(-1 < Z < 2)? Use the normal tables. Is this close to the area you estimated?

Answer 1