Let X be a Gaussian random variable with mean 0 and variance o = 1. We...

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Let X be a Gaussian random variable with mean 0 and variance o² = 1. We define three new random variables as Y = aX+b, Z = X², and W X³. We require that a 0. For reference, E[X³] = 0, E[X4] = 304, E[X³] = 0, and E[X6] = 150. = (a) Find the correlation coefficients for each pair (X, Y), (X, Z), and (X, W). (b) For the case of a = 5 and b = 0, randomly sample X 1000 times, use each X to get Y, and use a scatter plot to plot (X, Y) where the values of X is along the x-axis and the values of Y are along the y-axis. Describe how the correlation coefficient relates to what you see in the scatter plot. (c) For the case of a = -4 and b = 0, randomly sample X 1000 times, use each X to get Y, and use a scatter plot to plot (X, Y) where the values of X is along the x-axis and the values of Y are along the y-axis. Describe how the correlation coefficient relates to what you see in the scatter plot and explain the difference to part (b). (d) Now, randomly sample X 1000 times, use each X to get Z, and use a scatter plot to plot (X, Z) where the values of X is along the x-axis and the values of Z are along the y-axis. Describe how the correlation coefficient relates to what you see in the scatter plot and explain the difference to parts (b) and (c). (e) Now, randomly sample X 1000 times, use each X to get W, and use a scatter plot to plot (X, W) where the values of X is along the x-axis and the values of W are along the y-axis. Describe how the correlation coefficient relates to what you see in the scatter plot. Let X be a Gaussian random variable with mean 0 and variance o² = 1. We define three new random variables as Y = aX+b, Z = X², and W X³. We require that a 0. For reference, E[X³] = 0, E[X4] = 304, E[X³] = 0, and E[X6] = 150. = (a) Find the correlation coefficients for each pair (X, Y), (X, Z), and (X, W). (b) For the case of a = 5 and b = 0, randomly sample X 1000 times, use each X to get Y, and use a scatter plot to plot (X, Y) where the values of X is along the x-axis and the values of Y are along the y-axis. Describe how the correlation coefficient relates to what you see in the scatter plot. (c) For the case of a = -4 and b = 0, randomly sample X 1000 times, use each X to get Y, and use a scatter plot to plot (X, Y) where the values of X is along the x-axis and the values of Y are along the y-axis. Describe how the correlation coefficient relates to what you see in the scatter plot and explain the difference to part (b). (d) Now, randomly sample X 1000 times, use each X to get Z, and use a scatter plot to plot (X, Z) where the values of X is along the x-axis and the values of Z are along the y-axis. Describe how the correlation coefficient relates to what you see in the scatter plot and explain the difference to parts (b) and (c). (e) Now, randomly sample X 1000 times, use each X to get W, and use a scatter plot to plot (X, W) where the values of X is along the x-axis and the values of W are along the y-axis. Describe how the correlation coefficient relates to what you see in the scatter plot.