Find the mean and variance of the discrete random variable x . Coin Toss x is the number

Find the mean and variance of the discrete random variable x...

Key Concepts

Discrete Random Variables

Discrete random variables can assume only a countable number of distinct values. They are commonly used to model events with a finite or countable list of outcomes, such as the number of heads in coin tosses.

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent and identically distributed Bernoulli trials. It is especially applicable when each trial has two possible outcomes (success or failure) and a constant probability of success.

Expected Value (Mean)

The expected value of a random variable is a measure of its central tendency or long-term average outcome. For a binomial distribution, the mean is calculated as the product of the number of trials and the probability of success in each trial.

Variance

Variance quantifies the spread or dispersion of a random variable’s possible values around its expected value. In the context of a binomial distribution, the variance is given by the product of the number of trials, the probability of success, and the probability of failure.

Transcript

00:01 In the current problem, we have to find the expectation of x and then the variance of x, where x is the number of heads when a coin is tossed four times.

00:18 Okay, four times.

00:21 So we first should find out the sample space and we know that goes something like 8h, 8 ,000, then h, h, h, h -h -t and then h -h -h -t -h and h -h -t -t -t.

00:44 And then again it will start with so we did for h -h -h -t -t -t -t now we will change this part so it will start with h -t -t then again h -h -h then h -t h t and then again h t t and then h and then h t t and then now we have to change the first position so this entire thing will be repeated so t h h and then t h and then t h h and then t h h and then t th t h t h and then t h t t t t h and then t t t h and then t t t t h and t t t t t t t t t t t t t t t t t t now we have to write or find the number of heads so it's four over here three three and again over here three 2 2 1 again 3 2 2 1 2 1 2 1 0 then if we try to draw the calculation table it will be x x will be the number of heads so it will vary from 0 1 2 2 3 and 4 we know the probability of x now what is the total number of outcomes it is 4 to the 2 to the bar 4 it's 16 right why is the 2 to the bar 4 because there are 4 places and each has head or 10 two options so 2 into 2 in 2 oops i mean yeah 2 into 2 into 2 into 2 that's idea right so now how many times 0 heads occurring only one time how many times 4 heads are occurring? it is only one time so 1 by 16 each we can definitely be sure of it and write and then you can see a symmetry how many times out of this it occurred one okay there is there are total 16 cases right so in the four places how many ways you can have one eight and other three things so it is four see one ways right so it should be four ways 4 by 16.

03:42 Now let's verify from our calculation.

03:45 So if you see how many times one is occurring, we see a one here, you see another one here, you see another one here, and we see another one here.

03:54 So whatever we calculated or whatever a logic is is valid.

03:58 The same way for three also, those many times three also will occur.

04:04 So it is 4 by 16.

04:05 Why think on the same line how many ways can be this arrange 3 h s and 1 t where there are 4 places now of 4 we can choose 3 places so 4 c 3 and again 4 c 3 is 4 c1 hence 4 and let's verify that number over here how many times 3 is occurring 1 2 3 and 4 nobody else right so now what about 2 so when it is 2 it is head head and tail tail so out of four places how many ways we can choose two twos that is 4 c2 4c2 is nothing but factorial 4 divided by factorial 2 factorial 2 which is 6 ways correct so it is 6 by 16 okay so let's check that number 2 how many times 2 is occurring in the table if you see 1 2 2 3 3 4 4, 5, 6.

05:18 That means our calculation or any of our approaches are correct.

05:22 Now to find expectation x, it is x into px simple.

05:28 Now we have to do a multiplication.

05:31 This into this would give us 0.

05:33 This into this will be 4 by 16 itself.

05:37 This will be 12 by 16.

05:40 This will be 12 by 16 again.

05:43 And this will be 4...