The Poisson Distribution

Find $P(X \geq 3)$.

Find the probability that the sum of three independent values of $X$ is between 3 and 5 inclusive.

The sum of a large number, $n$, of values of $X$ is denoted by $T$. Using a suitable approximation, it was found that $P(T > 330) = 0.0391$, correct to 3 significant figures.

Find the value of $n$.

Find $\text{P}(3 \leqslant W + X \leqslant 5)$.

The random variable $S$ is the sum of $100$ independent values of $W$ and $200$ independent values of $X$.

Use a suitable approximation to find $\text{P}(S > 600)$.

The random variable $Y$ has the distribution $\text{Po}(\lambda)$, where $\lambda > 0$.

It is given that $\frac{5}{2}\text{P}(Y = 3) + \text{P}(Y = 4) = \text{P}(Y = 5)$.

Find the value of $\lambda$.

Write down an expression in terms of $e$ for $\text{P}(X = 12)$.

It is given that $\text{P}(X = n) = \text{P}(X = n + 1)$.

Write down an equation in $n$, and hence find the value of $n$.

Find $\text{P}(W \geq 4)$.

Two values of $W$ are chosen at random.

Find the probability that the sum of these two values is less than 3.

The random variable $X$ has the distribution $\text{Po}(200)$.

Use a suitable approximating distribution to find $\text{P}(X > 205)$.

Find the probability that, in a randomly chosen 1-minute period, at least 2 customers arrive.

State a suitable approximating distribution for $X$, giving the value(s) of any parameter(s).

Use your approximating distribution to find $P(20 < X < 30)$.

Find $P(S \le 3)$.

Show that the exact value of $\frac{P(S \le 2)}{P(S \le 1)}$ is $\frac{125}{44}$.

Find the probability that at least 4 cars and at least 2 trucks arrive at the fuel station during a randomly chosen 5-minute period.

Use a suitable approximating distribution to find the probability that a total of fewer than 145 cars and trucks arrive at the fuel station during a randomly chosen 2-hour period.

Find $P(2 < X < 5)$.

Find $P(X + Y > 2)$.

The total of 100 random values of $X$ and 150 random values of $Y$ is denoted by $T$.

Use a suitable approximating distribution to find $P(T < 560)$.

Use a suitable approximating distribution to find $P(X < 4)$.

The firm’s management wishes to decrease the value of $p$ by giving their employees some training. Their aim is that, for a data set containing 14500 characters, the value of $P(X = 0)$ for the new value of $p$ should be double the value of $P(X = 0)$ when $p = 0.0001$.

Use a suitable approximating distribution to find the new value of $p$.