Probability Distribution Coin Toss

Introduction to probability distribution coin toss:

In probability, a probability distribution used to identify with in a particular interval falling value of a probability. The probability distribution describes the range of the possible value of a random variable. The probability distribution is entirely described by the cumulative distribution function. In this article we shall discuss probability distribution coin toss example problem.

Probability Distribution Coin Toss Example Problem

Example: A coin is tossed for 65times. Find the expectation value of tail with the help of using the binomial distribution.

Solution:

Consider 65 times the coin is tossed, so n = 65

When toss a coin, we get probability of tail is p = 1/2

Probability of getting tail is represent by q

q = 1 – p                        [p + q = 1]

q = 1 – 1/2

q = 1/2

Expectation value:

μ = E[x] = np

= 65(1/2 )

= (65/2)

μ = E[x] = 32.5

Answer:

Expectation value = μ = E[x] = 32.5

Example 2: An unbiased die is rolling for 45 times. By using the binomial distribution find the expectation value of getting four?

Solution:

Let p be probability of getting success

Let q = 1 – p probability of getting failure

Let n be number of times the dice is thrown

An unbiased dice is rolling for 45 times, the value of n = 43

When we throe a unbiased dice, probability of getting four is p = 1/6

Probability of not getting four is represent by q

q = 1 – p

= 1 – 1/6

= 6/6– 1/6

= 6-1/6

q = 5/6

Expectation value:

μ = E(x) = np

= 45 (1/6)

= 45/6

= 7.5

μ = E(x) = 7.5

Answer:

Expectation value = μ = E[x] = 7.5

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Probability Distribution Coin Toss Practice Problem

Problem:

A coin is tossed for 42 times. Find the expectation value of tail with the help of using the binomial distribution.

Answer:

Expectation value = 21

Problem:

An unbiased die is rolling for 35 times. By using the binomial distribution find the expectation value of getting four?

Answer:

Expectation value = 5.8

Mixed Probability Distribution

Introduction to mixed probability distribution:

In this article we shall discuss the mixed probability distributions. Probability plays the main part in the mathematics and in all new fields. Probability is used to discover number of incidence of an event out of total outcomes. The probability of certain two random variables X and Y define on the equivalent probability space, the mixed distribution for X and Y defines the probability of events distinct in expressions of together X and Y.

Basic for Mixed Probability Distribution:

We found with a random test among probability measure P on a primary sample space. Now, we will confer two "mixed" cases for the probability distribution of a random variable in this case where the distribution is partly discrete and partly continuous and the case where the variable has both discrete coordinates and continuous coordinates.

Mixed Probability Distributions Consist of Following Cases:

Cumulative distribution case:

The cumulative distribution function for a couple of random variables is distinct in terms of their mixed probability distribution,

`F(x,y)=P(Xlt=x,Ylt=y)` .

Discrete distribution case:

For discrete random variables, the mixed probability distribution is,

`P(X=x and Y=y)=P(Y=y|X=x).P(X=x)` .

`=P(X=x|Y=y).P(Y=y)` .

Since these are probabilities, we have

`sum_(x)` `sum_(y)`` P(X=x and Y=y)=1` .

Continuous distribution case:

Likewise for continuous random variables, the mixed probability distribution function can be denoted as fX,Y(x, y) and this is,

`f_(X.Y)(x,y)=f_(Y|X)(y|x)f_(X)(x)=f_(X|Y)(x|y)f_(Y)(y)` .

Anywhere fY|X(y|x) and fX|Y(x|y) provide the conditional distribution of Y known X = x and of X known Y = y in that order, and fX(x) and fY(y) give the secondary distribution for X and Y correspondingly. Is this topic math 4th grade word problems hard for you? Watch out for my coming posts.

Again, since these are probability distributions, one has

`int_(x)int_(y)f_(X,Y)(x,y)dydx=1` .

Mixed probability distribution case:

Most of time X is continuous but Y is discrete. Intended for case, in a logistic regression, one might wish to predict the probability of a binary outcome Y conditional on the value of a continuously-distributed X. On the other hand, a "mixed probability distribution" can be distinct in together of two ways.

`f_(X,Y)(x,y)=f_(X|Y)(x|y)P(Y=y)` .

`=P(Y=y|X=x)f_(X)(x)` .

Properly, fX,Y(x, y) is the probability mass function of (X, Y) with value to the product measure on the particular supports of X and Y. each of these two decompositions can then be used to pick up the varied cumulative distribution function.

`F_(X,Y)(x,y)=sum_(tlt=y)int_(s=-oo)^x f_(X,Y)(s,t)ds` .

Grouping of arbitrary statistics of discrete and continuous random variables.