Let us discuss some properties of expected value. Let me recall that expected value for random variable with given probability distribution. Expected value for random variable that takes values X_1, X_2, X_3, and so on, X_n with probabilities P_1, P_2, P_3, and so on P_n is defined in the following way. Expected value of X is a sum over all I's from one to n P_i times X_i. So it means that I have to multiply these values by this probabilities and do a summation. Now, let us assume that I have new variable Z which is called to X plus C. What can you say about expected value of Z, or which is the same as expected value of X plus C? Let us continue this distribution for Z. The corresponding values are equal to X_1 plus C, X_2 plus C and so on X_n plus C. So it means that in this expected value we can write it in the following way. It is the sum of I from one to n X_i plus C times P_i. Then we can do basic arithmetic and get the following result. This is a sum and now we can expand this product and have X_i, P_i plus CP_i. Now we can expand it into two sums. The first sum, we see that this sum is equal to the expected value and the second sum, we have this C it does not depend on I so we can move it out of the summation sign. So it means that we have the following equation. Note that this sum is equal to one because it is the sum of all possible probabilities and it means that this thing is equal to EX plus C. Let me put brackets to make it more clear. Now we see that expected value of X plus C is expected value of X plus C. So we can take this constant out of the sign of expected value. This is first important property of expected value. Now let us consider a second property. How expected value behaves with multiplication by a constant? We can think about the following thing. What can we say about expected value of CX. Again, we can make X to be defined by probability distribution. Now we can see there are new variable CX and the values of CX are equal to C multiplied by the corresponding values of X. So we have CX_1 here, CX_2 here and so on CX_n here. Let us calculate the expected value. It is equal to the sum over I from one to n CX_i times P_i. It means that again we can move this C out of the sum and have the following formula. We see here an expected value of X. So we can conclude with the following, so we see that we can take constant out of the sign of expected value. To conclude, let me write these two important formulas once more. Expected value of X plus C is equal to expected value of X plus C and expected value of CX is equal to C expected value of X. These two properties are important and we will use them many times. We'll begin just try it now, we will use these formulas to prove a very nice property of random variable with symmetric probability mass function.