Week 4: Mathematical expectation

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Mean value of the hypergeometric distribution

The mean value of the hypergeometric distribution with parameters $(N,K,n)$ is equal to

$$\mu=\frac{nK}{N}.$$

As we discussed in class, the mean value was in a sense the center of the histogram: the point around which most of the histogram was concentrated. This is very vividly observed for the hypergeometric distribution. The plot bellow shows the mean value in blue for different values of $(N,K,n)$.

# nbi:hide_in
import matplotlib.pyplot as plt
import numpy as np
from scipy.special import comb
from ipywidgets import interact, IntSlider

def hypergeometric_pmf(N=80,K=40,n=30):
    range_x = np.arange(max(0, n-(N-K)), min(n, K)+1, 1)

    def hyper_pmf(N,K,n,i):
        pmf_val = comb(K, i, exact=True) * comb(N-K, n-i, exact=True) / comb(N, n, exact=True)
        return pmf_val
    mean = n * K / N

    pmf_values = np.array([hyper_pmf(N,K,n,i) for i in range_x])

    # plot setup
    plt.figure(figsize=(14,7)) 
    plt.axhline(y=0, color='k')
    plt.ylim(-0.02, max(np.max(pmf_values)+0.05, 0.2))
    plt.xlim(-2,N+2)
    plt.xticks(np.arange(0, N+1, 5))

    # PLotting with plt.bar instead of plt.hist works better when f(x) are knowwn
    plt.scatter(np.array([mean]),np.zeros(1), color ="blue", s=200)
    plt.scatter(range_x,np.zeros(range_x.shape), color ="red", s=20)
    plt.bar(range_x, pmf_values, width=1, color=(0.1, 0.1, 1, 0.1), edgecolor='blue', linewidth=1.3)
    plt.title("Hypergeometric distribution")
    plt.figtext(0.75,0.8, " N={} \n K={} \n n={} \n expectation={:.2f}".format(N,K,n, mean), ha="left", va="top",
            backgroundcolor=(0.1, 0.1, 1, 0.15), fontsize="large")
    plt.plot();

# create interactive variables
N = IntSlider(min=1.0, max=100.0, step=1.0, value=80, readout_format='')
K = IntSlider(min=1.0, max=N.value, step=1.0, value=40, readout_format='')
n = IntSlider(min=1.0, max=N.value, step=1.0, value=30, readout_format='')

# enforce K<=N and n<=N
def update_range(*args):
    K.max = N.value
    n.max = N.value
N.observe(update_range, 'value')

# display the interactive plot
interact(hypergeometric_pmf, N=N, K=K, n=n);

Geometric distribution

A random trial has a probability of success equal to $p$ and probability of failure $q=1-p$.Consider the following experiment: we are doing consecutive random trials until we reach a success.The set of outcomes has the form $s=\overbrace{FF\cdots F}^{n-1} S$ where number of $F$'s can be any number $n=1,2,\dots$. Due to independence, $P(s)=q^np$.

Let $X(s)$ denote the number of trials it took to reach success

$$X(\overbrace{FF\cdots F}^{n-1} S)=n.$$

Observe that

$$f(n)=q^{n-1}p,\quad n=1,2,\dots.$$

Definition (Geometric distribution)

The pmf of the random variable $X$ is called Geometric distribution. 
# nbi:hide_in
import matplotlib.pyplot as plt
import numpy as np
from ipywidgets import interact, FloatSlider


def geometric_pmf(p=0.5):
    q = 1-p
    N=50
    range_x = np.arange(1, N, 1)

    def geo_pmf(n):
        pmf_val = q**(n-1)*p
        return pmf_val
    mean = 1/p

    pmf_values = np.array([geo_pmf(n) for n in range_x])

    # plot setup
    plt.figure(figsize=(14,7)) 
    plt.axhline(y=0, color='k')
    plt.ylim(-0.02, max(np.max(pmf_values)+0.05, 0.2))
    plt.xlim(0, N+1)
    plt.xticks(np.arange(0, N+1, 5))

    # PLotting with plt.bar instead of plt.hist works better when f(x) are knowwn
    plt.scatter(np.array([mean]),np.zeros(1), color ="blue", s=200)
    plt.scatter(range_x,np.zeros(range_x.shape), color ="red", s=20)
    plt.bar(range_x, pmf_values, width=1, color=(0.1, 0.1, 1, 0.1), edgecolor='blue', linewidth=1.3)
    plt.title("Geometric distribution")
    plt.figtext(0.8,0.8, "p={}".format(p), ha="left", va="top",
            backgroundcolor=(0.1, 0.1, 1, 0.15), fontsize="large")
    plt.plot();

# create interactive variables
p = FloatSlider(min=0.01, max=1, step=0.01, value=0.5, readout_format='')

# display the interactive plot
interact(geometric_pmf, p=p);

Mean and Variance under linear transformation

Theorem

Suppose $X$ is a random variable with mean $\mu_X$ and standard deviation $\sigma_Y$. Let $Y=aX+b$ where $a$ and $b$ are any two numbers. Then mean and the standard deviation of $Y$ are $$\mu_Y=a\mu_X+b,\quad \sigma_Y=|a|\sigma_X.$$

Proof $\mu_Y=E[aX+b]=aE[X]+b=a\mu_X+b.$ Notice that

$$\text{Var}(Y)=E[(Y-E[Y])^2]=E[(aX+b-aE[X]-b)^2]=E[(aX-aE[X])^2]=a^2\text{Var}(X).$$

Taking square roots, we have $\sigma_Y=|a|\sigma_X$. $\blacksquare$

Example

Let $X$ be a random variable with $\text{range}(X)=\{-2,-1,0,1,2\}$ and

$$f_X(-2)=0.4, \quad f_X(-1)=0.25, \quad f_X(0)=0.15, \quad f_X(1)=0.1, \quad f_X(2)=0.1.$$

Take the random variable $Y=2X+1$.Notice that $\text{range}(Y)=\{-3,-1,1,3,5\}$ and

$$_Y(-3)=0.4, \quad f_Y(-1)=0.25, \quad f_Y(1)=0.15, \quad f_Y(3)=0.1, \quad f_Y(5)=0.1.$$

It can be checked that

$$\mu_X=-0.75, \quad \sigma_X\approx 1.414$$

and thus

$$\mu_Y=-0.5, \quad \sigma_Y\approx 2.828. $$ 
# nbi:hide_in
import matplotlib.pyplot as plt
import numpy as np

# values
range_x = np.array([-2,-1,0,1,2])
range_y = np.array([-3,-1,1,3,5])
xpmf_values = np.array([0.4, 0.25, 0.15, 0.1, 0.1])
ypmf_values = np.array([0.4, 0.25, 0.15, 0.1, 0.1])

# mean  
mean_x = np.average(range_x, weights = xpmf_values)
mean_y = np.average(range_y, weights = ypmf_values)

# variance
mean_x2 = np.average(np.power(range_x,2), weights = xpmf_values)
mean_y2 = np.average(np.power(range_y,2), weights = ypmf_values)
var_x = mean_x2 - mean_x**2
var_y = mean_y2 - mean_y**2

# set up the figure
fig, [ax1, ax2] = plt.subplots(1,2, num=1, clear=True)
plt.figure(figsize=(20,10)) 

ax1.set_ylim(-0.01,0.5) 
ax1.set_xlim(-5, 10)
ax1.axhline(y=0, color='k')
ax1.set_xticks(range_x)
ax1.set_yticks(xpmf_values)

ax2.set_ylim(-0.01, 0.5) 
ax2.set_xlim(-5, 10)
ax2.axhline(y=0, color='k')
ax2.set_xticks(range_y)
ax2.set_yticks(ypmf_values)


# 
ax1.scatter(np.array([mean_x]),np.zeros(1), color ="blue", s=200)
ax1.scatter(range_x,np.zeros(range_x.shape), color ="red", s=20)
ax1.bar(range_x, xpmf_values, width=1, color=(0.1, 0.1, 1, 0.1), edgecolor='blue', linewidth=1.3)
ax1.set_title(r"$\mu_X=${:.2f}, $\sigma^2_X\approx${:.3f}".format(mean_x, var_x))


# 
ax2.scatter(np.array([mean_y]),np.zeros(1), color ="blue", s=200)
ax2.scatter(range_y,np.zeros(range_y.shape), color ="red", s=20)
ax2.bar(range_y, ypmf_values, width=1, color=(0.1, 0.1, 1, 0.1), edgecolor='blue', linewidth=1.3)
ax2.set_title(r"$\mu_Y=${:.2f}, $\sigma^2_Y\approx${:.3f}".format(mean_y, var_y))

plt.show();

<Figure size 1440x720 with 0 Axes>

Bernoulli trials

Definition

In Bernoulli experiment the outcome is either of Type I (success) with probability $p$ or Type II (fail) with probability $q=1-p$.

A Bernoulli trials are the outcomes of several independent Bernoulli experiments with the same success probabilities.

Example Consider the coin flip experiment with a biased coin that lands head with probability $p$ and tail with probability $q=1-p$.A single coin flip will be a Bernoulli experiment.$n$ independent coin flips will be a Bernoulli trial.

Binomial distribution

Definition

Let $X$ denote the number of successes in $n$ Bernoulli trials with success probability $p$. Then we say that $X$ has the \textbf{binomial distribution} with parameters $(n,p)$.

If $X$ has binomial distribution with parameters $(n,p)$ then

  • $\text{range}(X)=\{0,1,\dots, n\}$.
  • $$f(x)={n\choose x}p^xq^{(n-x)}$$ ${n\choose x}$ ways there can be $x$ successes: the rest $n-x$ are failures.
# nbi:hide_in
import matplotlib.pyplot as plt
import numpy as np
from ipywidgets import interact, FloatSlider, IntSlider
from scipy.special import comb

def binomial_pmf(n, p):
    q = 1-p
    range_x = np.arange(0, n, 1)

    def binom_pmf(x):
        pmf_val = comb(n, x, exact=True) * p**x * q**(n-x)
        return pmf_val
    mean = n*p

    pmf_values = np.array([binom_pmf(x) for x in range_x])
    
    # plot setup
    plt.figure(figsize=(14,7)) 
    plt.axhline(y=0, color='k')
    plt.ylim(-0.02, max(np.max(pmf_values)+0.05, 0.2))
    plt.xlim(-1, n+1)
    plt.xticks(np.arange(0, n+1, 5))

    # PLotting with plt.bar instead of plt.hist works better when f(x) are knowwn
    plt.scatter(np.array([mean]),np.zeros(1), color ="blue", s=200)
    plt.scatter(range_x,np.zeros(range_x.shape), color ="red", s=20)
    plt.bar(range_x, pmf_values, width=1, color=(0.1, 0.1, 1, 0.1), edgecolor='blue', linewidth=1.3)
    #plt.title("Binomial distribution")
    plt.figtext(0.8,0.8, " n={} \n p={}".format(n, p), ha="left", va="top",
                backgroundcolor=(0.1, 0.1, 1, 0.15), fontsize="large")
    plt.plot();

# create interactive variables
p = FloatSlider(min=0.01, max=1, step=0.01, value=0.5, readout_format='')
n = IntSlider(min=1, max=100, step=1, value=50, readout_format='')
# display the interactive plot
interact(binomial_pmf, n=n, p=p);

Binomial vs hypergeometric

In the hypergeometric distribution with paramaters $(N,K,n)$ when $N>>n$, it is very close to the binomial distribution with paameters $(n,\frac{K}{N})$. In the plot below you can increase the value of the $N$ while leaving $n$ fixed to see that their histograms approach one another.

# nbi:hide_in
import matplotlib.pyplot as plt
import numpy as np
from scipy.special import comb
from ipywidgets import interact, IntSlider

def hypergeo_binom_pmf(N=80,K=40,n=30):
    p = K/N
    q = 1-p
    brange_x = np.arange(0, n, 1)
    hrange_x = np.arange(max(0, n-(N-K)), min(n, K)+1, 1)

    def hyper_pmf(N,K,n,i):
        pmf_val = comb(K, i, exact=True) * comb(N-K, n-i, exact=True) / comb(N, n, exact=True)
        return pmf_val
    mean = n * K / N

    hpmf_values = np.array([hyper_pmf(N,K,n,i) for i in hrange_x])
    
    def binom_pmf(x):
        pmf_val = comb(n, x, exact=True) * p**x * q**(n-x)
        return pmf_val
    mean = n*p

    bpmf_values = np.array([binom_pmf(x) for x in brange_x])

    # plot setup
    plt.figure(figsize=(14,7)) 
    plt.axhline(y=0, color='k')
    plt.ylim(-0.02, max(np.max(hpmf_values)+0.05, np.max(bpmf_values)+0.05, 0.2))
    plt.xlim(-2,max(np.max(hrange_x), np.max(brange_x))+2)

    # Plotting hypergeometric
    plt.bar(hrange_x, hpmf_values, width=1, color=(0.1, 0.1, 1, 0.1), edgecolor='blue', 
            linewidth=1.3, label="Hypergeometric")
    
    # Plotting binomial
    plt.bar(brange_x, bpmf_values, width=1, color=(0.3, 0.5, 0.3, 0.1), edgecolor='green',
            linewidth=1.3, label="Binomial")
    plt.figtext(0.78,0.75, " N={} \n K={} \n n={}".format(N,K,n), ha="left", va="top",
            backgroundcolor=(0.1, 0.1, 1, 0.15), fontsize="large")
    plt.legend()
    plt.plot();

# create interactive variables
N = IntSlider(min=1.0, max=200.0, step=1.0, value=300, readout_format='')
K = IntSlider(min=1.0, max=N.value, step=1.0, value=50, readout_format='')
n = IntSlider(min=1.0, max=N.value, step=1.0, value=20, readout_format='')

# enforce K<=N and n<=N
def update_range(*args):
    K.max = N.value
    n.max = N.value
N.observe(update_range, 'value')

# display the interactive plot
interact(hypergeo_binom_pmf, N=N, K=K, n=n);

cdf of the binomial distribution

The cdf of the binomial distribution can be computed as $$F(x)=E[X\leq x]=\sum_{y=0}^{\lfloor x\rfloor}{n\choose x}p^xq^{(n-x)}\text{ for } x\in [0,n].$$

However, $F(x)$ does not have a simple closed-form formula. In old times people would use statistical tables that show precomputed values of the cdf for different choices of parameters $(n,p)$. Nowdays, computers can give you very precise values for the cdf for any $n,p,x$.

In the plot below, the shaded area is the value of the cdf at point $x$. The point $x$ is the black dot on the plot. The value of the cdf up to 4 digit precision is given at the top of the plot.

# nbi:hide_in
import matplotlib.pyplot as plt
import numpy as np
from ipywidgets import interact, FloatSlider, IntSlider
from scipy.special import comb
from scipy.stats import binom

def binomial_cdf(n, p, x):
    q = 1-p
    range_x = np.arange(0, n, 1)

    def binom_pmf(x):
        pmf_val = comb(n, x, exact=True) * p**x * q**(n-x)
        return pmf_val
    mean = n*p

    pmf_values = np.array([binom_pmf(x) for x in range_x])
    
    # plot setup
    plt.figure(figsize=(20,10)) 
    plt.axhline(y=0, color='k')
    plt.ylim(-0.02, max(np.max(pmf_values)+0.05, 0.2))
    plt.xlim(-4, n+4)

    # PLotting with plt.bar instead of plt.hist works better when f(x) are knowwn
    plt.scatter(x,np.zeros(1), color ="black", s=100)
    plt.scatter(range_x,np.zeros(range_x.shape), color ="red", s=20)
    barlist = plt.bar(range_x, pmf_values, width=1, color=(1, 1, 1, 0), edgecolor='blue', linewidth=1.3, label="Histogran")
    for ind, val in enumerate(range_x):
        if val<=x:
            barlist[ind].set_color((0.1, 0.1, 1, 0.15))
            barlist[ind].set_edgecolor("blue")    
    cdf = binom.cdf(x, n, p)
    plt.title("cdf and pdf of binomial distribution" )
    plt.figtext(0.8,0.8, " n={} \n p={} \n x={} \n cdf = {:.4f}".format(n, p, x, cdf), ha="left", va="top",
                backgroundcolor=(0.1, 0.1, 1, 0.15), fontsize="large")
    plt.plot();

# create interactive variables
p = FloatSlider(min=0.01, max=1, step=0.01, value=0.25, readout_format='')
n = IntSlider(min=1, max=100, step=1, value=3, readout_format='')
x = FloatSlider(min=-4, max=10, step=0.1, value=1.75, readout_format='')

# enforce values for x
def update_range(*args):
    x.max = max(n.value+4,10)
n.observe(update_range, 'value')

# display the interactive plot
interact(binomial_cdf, n=n, p=p, x=x);

Negative binomial distribution

Definition

Let $X$ be the number of independent Bernoulli trials with success probability $p$ needed until exactly r successes occur. Then we say $X$ has the **negative binomial distribution** with parameters $(r,p)$.
  • $\text{range}(X)=\{r,r+1,\dots\}.$ *If $X(s)=x$, means there are $r-1$ successes and $x-r$ fails in the first $x-1$ trials and the $x-th $ trial is a success. Therefore
$$f(x)={x-1\choose r-1}p^{r}q^{x-r}.$$
  • When $r=1$, the negative binomial distribution is the same as the geometric distribution. 
# nbi:hide_in
import matplotlib.pyplot as plt
import numpy as np
from ipywidgets import interact, FloatSlider


def negbinomial_pmf(r, p):
    q = 1-p
    N=100
    range_x = np.arange(r, N, 1)

    def negbin_pmf(x):
        pmf_val = comb(x-1, r-1, exact=True) * p**r * q**(x-r)
        return pmf_val
    mean = r/p

    pmf_values = np.array([negbin_pmf(x) for x in range_x])

    # plot setup
    plt.figure(figsize=(14,7)) 
    plt.axhline(y=0, color='k')
    plt.ylim(-0.02, max(np.max(pmf_values)+0.05, 0.1))
    plt.xlim(np.min(range_x)-2, N+1)

    # PLotting with plt.bar instead of plt.hist works better when f(x) are knowwn
    plt.scatter(np.array([mean]),np.zeros(1), color ="blue", s=200)
    plt.scatter(range_x,np.zeros(range_x.shape), color ="red", s=20)
    plt.bar(range_x, pmf_values, width=1, color=(0.1, 0.1, 1, 0.1), edgecolor='blue', linewidth=1.3)
    plt.title("Negative binomial distribution")
    plt.figtext(0.8,0.8, " r={} \n p={} ".format(r,p), ha="left", va="top",
            backgroundcolor=(0.1, 0.1, 1, 0.15), fontsize="large")
    plt.plot();

# create interactive variables
r = IntSlider(min=1, max=50, step=1, value=30, readout_format='')
p = FloatSlider(min=0.01, max=1, step=0.01, value=0.5, readout_format='')

# display the interactive plot
interact(negbinomial_pmf, r=r, p=p);

Poisson distribution

Definition

A discrete random variable has Poisson distribution with parameter $\lambda>0$ if $$\text{range}(X)=\{0,1,2,3,\dots\}$$ and $$f(x)=\frac{e^{-\lambda}\lambda^x}{x!},\quad x\in\text{range}(X).$$ $\lambda$ is called the **expected rate**.

Notice that $f(x)$ is indeed a pmf:

$$\sum_{x\in \text{range}(X)}f(x)=\sum_{x=0}^\infty \frac{e^{-\lambda}\lambda^x}{x!}=e^{-\lambda} \sum_{x=0}^\infty \frac{\lambda^x}{x!}=e^{-\lambda}e^{\lambda}=1.$$ 
# nbi:hide_in
import matplotlib.pyplot as plt
import numpy as np
from scipy.special import factorial
from ipywidgets import interact, FloatSlider


def poisson_pmf(lam):
    N=50
    range_x = np.arange(0, N, 1)

    def poiss_pmf(x):
        pmf_val = np.exp(-lam) * np.power(lam, x) / factorial(x)
        return pmf_val
    mean = lam

    pmf_values = np.array([poiss_pmf(x) for x in range_x])

    # plot setup
    plt.figure(figsize=(14,7)) 
    plt.axhline(y=0, color='k')
    plt.ylim(-0.02, max(np.max(pmf_values)+0.05, 0.1))
    plt.xlim(0, N+1)
    plt.xticks(np.arange(0, N+1, 5))

    # PLotting with plt.bar instead of plt.hist works better when f(x) are knowwn
    plt.scatter(np.array([mean]),np.zeros(1), color ="blue", s=200)
    plt.scatter(range_x,np.zeros(range_x.shape), color ="red", s=20)
    plt.bar(range_x, pmf_values, width=1, color=(0.1, 0.1, 1, 0.1), edgecolor='blue', linewidth=1.3)
    plt.title("Geometric distribution")
    plt.figtext(0.8,0.8, r"$\lambda$={}".format(lam), ha="left", va="top",
            backgroundcolor=(0.1, 0.1, 1, 0.15), fontsize="large")
    plt.plot();

# create interactive variables
lam = FloatSlider(min=0.0, max=100, step=0.5, value=30, readout_format='', description=r"$\lambda$")

# display the interactive plot
interact(poisson_pmf, lam=lam);

Motivation

The Poisson distribution arises in the following way (made up scenario): grocery route bus can arrive every $\frac{1}{n}$ hour. It is a very unpredictable bus but we know that there are on average $\lambda$ buses arriving every hour. Assuming whether a bus arrives or not at given time is independent from other times, this becomes a binomial distribution with parameter $p=\frac{\lambda}{n}$. We know that the expect number of successes in a binomial distribution with parameters $(n,p)$ is $np$ and so $p=\frac{\lambda}{n}$. Therefore, the probability of $x$ number of buses arriving in an hour will be

$$f(x)={n\choose x}p^x(1-p)^{n-x}={n\choose x}\left(\frac{\lambda}{n}\right)^x\left(1-\frac{\lambda}{n}\right)^{n-x},\quad x=0,1,2,\dots, n. $$
  • Now imagine the bus starts arriving at more irregular times. To accommodate that we can increase the $n$.
  • When $n\to \infty$ that will correspond to the buses potentially arriving at any moment in the continuous interval.
  • In that case
$$f(x)=\lim_{n\to \infty}{n\choose x}\left(\frac{\lambda}{n}\right)^x\left(1-\frac{\lambda}{n}\right)^{n-x}, \quad x=0,1,2,\dots.$$

To compute this limit notice that

$$\lim_{n\to \infty}{n\choose x}\left(\frac{\lambda}{n}\right)^x=\frac{\lambda^x}{x!}\lim_{n\to \infty}\frac{n(n-1)\cdots (n-x+1)}{•n^x}=\frac{\lambda^x}{x!}$$

and (you should know this from calculus)

$$\lim_{n\to \infty}\left(1-\frac{\lambda}{n}\right)^{n-x}=e^{-\lambda}.$$

Consequently, if the buses can potentially arrive any moment within the hour, with $\lambda$ buses arriving in average at every hour, then the probability that $x$ buses will arrive in a given hour is $$f(x)=\frac{e^{-\lambda}\lambda^x}{x!},\quad x=0,1,2,\dots.$$

Let us see that the Poisson distribution with parameter $\lambda$ and the binomial distribution with parameters $(n,p=\frac{\lambda}{n})$ are close to each other when $n$ is large by looking at their histograms.

# nbi:hide_in
import matplotlib.pyplot as plt
import numpy as np
from scipy.special import comb
from ipywidgets import interact, IntSlider

def poiss_binom_pmf(n, lam):
    p = lam/n
    q = 1-p
    N = 50
    brange_x = np.arange(0, n, 1)
    prange_x = np.arange(0, N, 1)

    def poiss_pmf(x):
        pmf_val = np.exp(-lam) * np.power(lam, x) / factorial(x)
        return pmf_val
    mean = lam

    ppmf_values = np.array([poiss_pmf(x) for x in prange_x])
    
    def binom_pmf(x):
        pmf_val = comb(n, x, exact=True) * p**x * q**(n-x)
        return pmf_val
    mean = n*p

    bpmf_values = np.array([binom_pmf(x) for x in brange_x])

    # plot setup
    plt.figure(figsize=(14,7)) 
    plt.axhline(y=0, color='k')
    plt.xlim(-2,20)

    # Plotting hypergeometric
    plt.bar(prange_x, ppmf_values, width=1, color=(0.1, 0.1, 1, 0.1), edgecolor='blue', 
            linewidth=1.3, label="Poisson")
    
    # Plotting binomial
    plt.bar(brange_x, bpmf_values, width=1, color=(0.3, 0.5, 0.3, 0.1), edgecolor='green',
            linewidth=1.3, label="Binomial")
    plt.figtext(0.83,0.75, r" $\lambda$={}".format(lam)+"\n n = {}".format(n), ha="left", va="top",
            backgroundcolor=(0.1, 0.1, 1, 0.15), fontsize="large")
    plt.legend()
    plt.plot();

# create interactive variables
lam = FloatSlider(min=0.0, max=100, step=0.5, value=5, readout_format='', description=r"$\lambda$")
n = IntSlider(min=np.floor(lam.value)+1, max=200, step=1, value=100, readout_format='')

# enforce K<=N and n<=N
def update_range(*args):
    n.min = np.floor(lam.value)+1
    
lam.observe(update_range, 'value')

# display the interactive plot
interact(poiss_binom_pmf,  lam=lam, n=n);