 Welcome to the session. In this session we shall first discuss about a random variable. A random variable is a real valued function whose domain is the sample space of a random experiment. Random variable is usually denoted by let's consider an experiment of tossing a coin two times in succession. In this case the sample space s would be equal to h h h t th t t. Let x be the number of heads obtained. So this x is the random variable. Now we shall find the value of this random variable x for each outcome. Value of x for the outcome h h would be equal to 2 since we have two heads in this case and x is the number of heads obtained. Then x for the outcome h t is equal to 1 since we have one head in this case. x for the outcome th is also 1 since there is one head in this case. Then x for the outcome t t is 0 since there is no head in this case. More than one random variable can be defined on the same sample space. Next we shall discuss probability distribution of a random variable. A description giving the values of the random variable along with the corresponding properties is called the probability distribution of the random variable x. So we say the probability distribution of a random variable x is the system of numbers random variable x with the values x1, x2 and so on up to xn with probabilities given by px as p1, p2 and so on up to pn. Where we have pi is greater than 0 and sigma pi i goes from 1 to n is equal to 1. The real numbers x1, x2 and so on up to xn are the possible values of the random variable x and pi is the probability of the random variable x taking the value xi. That is we have probability of x equal to xi is equal to pi. Let's consider the same example as discussed above in which we had the sample space s equal to hh, ht, th, tt and x is the number of heads obtained. This is the experiment when we tossed a coin two times in succession. Since x is the number of heads so x would get the values 0, 1, 2. As you can see in the sample space either we have no heads or we have one head or we have two heads. So x would take the value 0, 1, 2. Now probability when number of heads is 0 that is x is equal to 0 that is probability of the outcome tt would be equal to 1 upon 4. Then probability x equal to 1 is equal to probability of the outcome ht or th. In this case the probability would be equal to 2 upon 4 that is 1 upon 2. Then next we have probability when x equal to 2 that is probability of the outcome hh. In this case the probability would be 1 upon 4. So the probability distribution of x is given by x equal to xi with values 0, 1, 2. Then pi that is probability for each xi would be 1 upon 4, 1 upon 2 and 1 upon 4. That is when we have xi equal to 0 its corresponding probability is 1 upon 4, xi equal to 1 corresponding probability is 1 upon 2, xi equal to 2 corresponding probability is 1 upon 4. As you can see each pi is greater than 0 and sigma pi equal to 1 upon 4 plus 1 upon 2 plus 1 upon 4 is equal to 1. Next we have mean of a random variable. Basically mean is a measure of location or central tendency in the sense that it roughly locates a middle or average value of the variable. Now let's define the mean of a random variable. Let x be a random variable whose possible values x1, x2 and so on up to xn occur with probabilities p2 and so on up to pn respectively. The mean of x that is mean of the random variable x is denoted by mu which is equal to summation xi pi i goes from 1 to n. Mean of a random variable x is also called the expectation of x which is denoted by ex this is equal to mu equal to summation xi pi i goes from 1 to n which can also be written as x1p1 plus x2p2 plus and so on up to xnpn. Let's consider the same example that we had considered above in which we tossed a coin two times in succession. In that case the possible values of x that we got were x1 equal to 0, x2 equal to 1, x3 equal to 2. Now the corresponding probabilities that is probability when x1 equal to 0 is given by p1 equal to 1 upon 4. When x2 equal to 1 its probability p2 is given by 1 upon 2. When x3 equal to 2 its probability is given by p3 equal to 1 upon 4. Now the mean of the random variable x is given by mu equal to summation xi pi i goes from 1 to 3 in this case that is equal to x1 that is 0 multiplied by p1 1 upon 4 plus x2 1 multiplied by p2 that is 1 upon 2 plus x3 that is 2 multiplied by p3 that is 1 upon 4. This comes out to be equal to 1 that is mean of the random variable x given by mu is equal to 1. Next we discuss variance of a random variable. Now the mean of a random variable does not give us information about the variability in the values of a random variable. So the variability or the spread in the values of a random variable may be measured by variance. Now let's define the variance of a random variable. Let x be a random variable whose possible values x1, x2 and so on up to xn occur with probabilities px1, px2 and so on up to pxn respectively. Then variance of x denoted by variance x or sigma x2 is given by summation xi minus mu the whole square multiplied by p of xi where i goes from 1 to n. Or equivalently this could be written as sigma x2 is equal to expectation of x minus mu the whole square. Again we shall consider the same example where we had x1 equal to 0, x2 equal to 1, x3 equal to 2 and the corresponding probability is given by px1 equal to 1 upon 4, px2 equal to 1 upon 2 and px3 equal to 1 upon 4. Now the variance of x that is sigma x2 is equal to summation xi minus mu the whole square into pxi i goes from 1 to 3 in this case. Here mu that is mean of the random variable x is 1 that we have already found out in the above example. So now sigma x2 that is variance of x is equal to x1 that is 0 minus mu that is 1 the whole square into px1 that is 1 upon 4 plus x2 that is 1 minus mu that is 1 the whole square into px2 that is 1 upon 2 plus x3 that is 2 minus mu that is 1 the whole square into px3 that is 1 upon 4. This comes out to be equal to 1 upon 2 that is variance of the random variable x that is sigma x2 is 1 upon 2. Now the non-negative number that is sigma x is equal to square root of variance of x that is square root of summation xi minus mu the whole square into pxi i goes from 1 to n. This is called the standard deviation of the random variable x that is the sigma x is the standard deviation. We have one more formula for the variance of x given by expectation of x square that is ex square minus expectation of x the whole square. From the above example we have variance of x that is sigma x square is 1 upon 2. So the standard deviation given by sigma x is equal to square root of 1 upon 2 that is equal to 1 upon square root 2 which is equal to 0.707 that is the standard deviation is given by 0.707. This completes the session hope you have understood what is a random variable probability distribution of a random variable and how we find mean variance and standard deviation of a random variable.