A discrete random variable Y has pmf (probability mass function) given by p=c/(y+4), y=-1,0,1,2, where c is a constant. Find the standard deviation of Y.

First, we need to find the constant c. The sum of all probabilities in a probability mass function must equal 1. So, we have:

c/(-1+4) + c/(0+4) + c/(1+4) + c/(2+4) = 1 c/3 + c/4 + c/5 + c/6 = 1 Multiply through by 60 (the least common multiple of 3, 4, 5, 6) to clear the fractions: 20c + 15c + 12c + 10c = 60 57c = 60 c = 60/57 = 20/19

Now, we can find the mean (expected value) of Y. The mean of a discrete random variable is given by the sum of the product of each outcome and its probability. So, we have:

E(Y) = Σ [y * p(y)] = -1 * (20/19)/3 + 0 * (20/19)/4 + 1 * (20/19)/5 + 2 * (20/19)/6 = -20/57 + 0 + 4/95 + 20/285 = -20/57 + 4/95 + 20/285 = -0.350877 + 0.042105 + 0.070175 = -0.238596

Next, we find the variance of Y. The variance of a discrete random variable is given by the sum of the square of the difference between each outcome and the mean, multiplied by the probability of each outcome. So, we have:

Var(Y) = Σ [(y - E(Y))^2 * p(y)] = (-1 - (-0.238596))^2 * (20/19)/3 + (0 - (-0.238596))^2 * (20/19)/4 + (1 - (-0.238596))^2 * (20/19)/5 + (2 - (-0.238596))^2 * (20/19)/6 = 0.5806 + 0.0113 + 0.1581 + 0.4743 = 1.2243

Finally, the standard deviation of Y is the square root of the variance. So, we have:

SD(Y) = sqrt(Var(Y)) = sqrt(1.2243) = 1.107

So, the standard deviation of Y is approximately 1.107.