# Cumulative distribution function of a discrete random variable

Can someone please explain how we got the pmf of the X ~ Bin (2, 2/3) ? (I understand that it is the Binomial distribution but where did we get the 0, 1, 2 from)

• Well, CDF of Binomial distribution shows the probability that the number of successes is less than or equal to a specific value. So, your CDF will show the probability that you have 0 successes, at most 1 success, and at most 2 successes (which, of course, covers all sample space). You can calculate all of it from the PMF you circled. Mar 1, 2020 at 1:06

The Binomial distribution is used to model the situation when we perform a trial a certain number of times. We call one outcome of the trial a 'success' and the alternative to this is a 'failure.' We then count the number of trials. The Binomial distribution gives us the probability of a particular number of successes.

Usually $$n$$ is the number of trials, $$p$$ is the probability of success, $$1-p$$ or $$q$$ is the probability of failure. $$r$$ is used for a particular number of successes.

Example: Janice plays 4 games of chess against Jack. She has a probability of 0.8 of winning a game.

In this example $$n=4$$ and $$p=0.8$$

Janice can win 0 games, 1 game, 2 games, 3 games or 4 games. These correspond to $$r=0, r=1, r=2, r=3, r=4$$.

The probability that Janice wins 3 games is given by $$P(X=3)$$. To find this we substitute $$n=4,p=0.8, r=3$$ into the formula $$P(X=r)={}_nC_rp^r(1-p)^{n-r}$$

$$P(X=3)={}_4C_30.8^30.2^1=0.4096$$

For $$X$$ ~ $$Bin(n,p)$$ we have $$P(X=r)={}_nC_rp^r(1-p)^{n-r}$$

In your case the possible values of $$r$$ are 0, 1, 2.

$$P(X=0)={}_2C_0(\frac23)^0(\frac13)^2=1 \times 1 \times \frac 19=\frac 19$$

$$P(X=1)={}_2C_1(\frac23)^1(\frac13)^1=2 \times \frac 23 \times \frac 13=\frac 49$$

$$P(X=2)={}_2C_2(\frac23)^2(\frac13)^0=1 \times \frac 49 \times 1=\frac 49$$

• Thank you for the explanation! But I still don't get why possible values for r are specifically from 0 to 2? Mar 1, 2020 at 1:07
• Thank you so much, now I understand it. How do we get the value of s then? 3/2 Mar 1, 2020 at 1:23
• You want to work out the value of $F$ for all values between 0 and $n$. They just picked $\frac 32$ as an example.
– tomi
Mar 1, 2020 at 1:28