The number of winter storms in a good year is a Poisson random variable with mean 3, whereas the number in a bad year is a Poisson random variable with mean 5. If next year will be a good year with probability .4 or a bad year with probability .6, find the expected value and variance of the number of storms that will occur.

Answer:  Mean = 4.8 and variance = 5.16Step-by-step explanation:Since we have given Let X be the number of storms occur in next yearY= 1 if the next year is good.Y=2 if the next year is bad.Mean for good year = 3probability for good year = 0.4Mean for bad year = 5probability for bad year = 0.6So, Expected value would be [tex]E[x]=\sum xp(x)\\\\=3\times 0.4+5\times 0.6\\\\=1.2+3\\=4.2[/tex]Variance of the number of storms that will occur.[tex]Var[x]=E[x^2]-(E[x])^2[/tex][tex]E[x^2]=E[x^2|Y=1].P(Y=1)+E[x^2|Y=2].P(Y=2)\\\\=(3+9)\times 0.4+(5+25)\times 0.6\\\\=12\times 0.4+30\times 0.6\\\\=4.8+18\\\\=22.8[/tex]So, Variance would be [tex]\sigma^2=22.8-(4.2)^2\\\\=5.16[/tex]Hence, Mean = 4.8 and variance = 5.16