It has been a bad day for the stock market and you have heard that only 30% of all stocks gained value.

 It has been a bad day for the stock market and you have heard that only 30% of all stocks gained value. Suppose you have a portfolio of 10 securities and assume a binomial distribution for the number of your stocks that gained value. (a) What assumptions are being made when you use the binomial distribution in this way? (b) Find the expectation and standard deviation of the number of securities in your portfolio that gained value. (c) What is the probability that six or fewer securities in your portfolio gained value? (d) Find the probability that exactly six securities in your portfolio gained value. (e) What is the probability that at least one of your securities gained value?